When good natured arguments and differences of view arise I tend to try to agree / disagree with what's been said, but only after consulting the oracle, which in the case of the English language means reference to the Oxford English Dictionary (OED).
Here's how it records the usage of Aeroplane, Airplane and Aerodrome.
Aeroplane
1. A flattened structure, originally plane but later aerodynamically curved, that forms the principal lifting surface of an aircraft; esp. a wing. Cf. AEROFOIL n. 1 , PLANE n.3 5. Now disused.
1866 F. H. WENHAM in 1st Ann. Rep. Aëronaut. Soc. 33 In the flying mechanism of beetles when the wing-cases are opened, they are checked by a stop, which sets them at a fixed angle. It is probable that these serve as ‘aeroplanes’, for carrying the weight of the insect. 1866 F. H. WENHAM in 1st Ann. Rep. Aëronaut. Soc. 37 The rude contrivance just described [sc. a glider] had taught, first, that the webs, or aeroplanes, must not be distended in a frame... A thin steel tie-band served as the foundation of the superposed aeroplanes. 1868 3rd Ann. Rep. Aëronaut. Soc. 36 He had turned his attention to the wing and to the sustainer, or what he might call the aëroplane. 1894 O. CHANUTE Progress in Flying Machines 237 This main aeroplane is trussed and stiffened in every direction by wire stays. 1905 G. BACON Balloons 111 What are called ‘aeroplanes’ large flat surfaces, light but rigid inclined at a suitable angle to the horizon. 1907 F. W. LANCHESTER Aerodynamics v. 179 The author does not employ the term aeroplane outside its correct signification, that is to say, to denote other than a true or plane aeroplane; the misuse of the word being avoided by the introduction of the word aerofoil, to denote a supporting member, or organ of sustentation of undefined form. 1910 R. FERRIS How it Flies xx. 454 Angle of Entry, the angle made by the tangent to the curve of the aeroplane surface at its forward edge, with the direction, or line, of travel.
2. Now chiefly Brit. An aircraft which relies on aerodynamic lift for flight; a heavier-than-air aircraft; esp. one having fixed wings and using propellers or jet engines to provide thrust. Cf. AIRPLANE n. 2b , PLANE n.5 , AIRCRAFT n.
In the period to 1900 when aircraft were still rudimentary in design and the word was still active in sense 1 the denotation is sometimes not clearly either ‘a surface’ or ‘an aircraft’, but something between the two concepts (see quots. 1868 , 18942 , 1896). For a full discussion of the development of the term in this period see S. Stubelius Airship, Aeroplane, Aircraft (1958 ) 251ff.
The equivalent term in North America is airplane.
1868 Eng. Mechanic 24 Apr. 91/2 We have yet to see the ‘aëroplane’ with buoyancy sufficient to sustain 150 lb., or with apparatus sufficiently light and portable to make headway on an ‘air plane’... Supposing an aëroplane to have raised itself, if it reared out of equilibium it and the occupant would come to grief. 1873 D. S. BROWN in 8th Ann. Rep. Aëronaut. Soc. 17, I think this [sc. impetus] will be more requisite with respect to the aëroplane than any other vehicle. 1873 Ann. Rep. Aëronaut. Soc. 20 Mr. Bennett introduced an Aëroplane invented by a Frenchman, to be worked by a screw by motive power derived from elastic springs. 1894 in Aeronaut. Ann. (1895) 152 With a dirigible aeroplane or soaring machine the rate of speed is practically a matter of choice. 1894 O. CHANUTE Progress in Flying Machines 72 It was not until 1842 that an aeroplane, as we now understand the term, consisting of planes to sustain the weight, and of a screw to propel, was first proposed. 1896 Westm. Gaz. 15 Sept. 2/1 Hargrave stands alone as one who has developed simultaneously the best form of aeroplane and motor before attempting to combine them in a flying-machine... Lilienthal appears to have confined himself entirely to practising with a motorless aeroplane formed of a double set of wings. 1905 W. WRIGHT Let. 28 Nov. in Papers Wilbur & Orville Wright (1953) I. 529 The first free flight through the air with a motor-driven aeroplane. 1908 Westm. Gaz. 11 May 4/1 The double box-kite aeroplane with which Mr. Farman won the Archdeacon Prize in Paris recently. 1934 O. WRIGHT Let. 10 Jan. in Papers W. & O. Wright (1953) II. 1162 The helicopter type of aeroplane offers several seemingly insurmountable difficulties. 1944 LD. ALANBROOKE Diary 17 Jan. in War Diaries (2001) 514 Latest reports on German pilotless aeroplanes. 1971 H. MACMILLAN Riding Storm iv. 98 Russian weapons were paraded through the streets, and Russian aeroplanes gave a display. 2005 BBC Focus Dec. 72/3 The mammoth aeroplane will hold nearly 600 people, and dwarfs even the biggest jumbo jet in the skies.
3. An airship provided with planes (PLANE n.3 5(a)). Obs. rare.
1884 Pall Mall Gaz. 28 Aug. 4/1 As soon as the Aero-Plane has been seen floating to and fro over the city of San Francisco, steered at pleasure this way and that, and carrying a number of passengers.
COMPOUNDS
General attrib., appositive, and objective, as, aeroplane carrier, aeroplane wing, etc.
1872 7th Ann. Rep. Aëronaut. Soc. 15 At an angle of ten degrees, about one man power would be sufficient to drive an aëroplane machine twenty miles an hour. 1872 7th Ann. Rep. Aëronaut. Soc. 17 An aëroplane, or rather the pair of aëroplane wings, must be long and narrow. 1896 H. S. MAXIM in Aeronaut. Ann. 2 38 The next machine was on the kite or aeroplane system. 1902 Aeronaut. World (U.S.) 1 Oct. 58/1 This bird-like aeroplane machine. 1920 Proc. Air Conference, London 96 Movement by sea is a slow business unless aeroplane carriers are available. 1923 R. KIPLING Irish Guards in Great War I. 50 ‘Aeroplane duty’ was another invention of those early days. A Company was told off daily to look out for aeroplanes. 1932 H. NICOLSON Public Faces xi. 301 She was an aeroplane carrier of His Majesty’s Navy. 1958 Spectator 10 Jan. 47/1 HQ Tank Corps arranged with the RFC to have daily aeroplane photographs taken of the front over which tanks would advance. 1974 Philos. Trans. (Royal Soc.) B. 267 583 The finger bone which forms the leading edge of the wing has exactly the transverse section as the corresponding part of an aeroplane wing. 1998 K. LETTE Altar Ego (1999) xxxi. 271 Treading in a puddle in the aeroplane toilet in absorbent airline socks.
DERIVATIVES
aeroplanist n. now rare a person who flies an aeroplane; a pilot.
1906 Daily Mail 26 Nov. 7/5 The first successful *aeroplanist in Great Britain will win as much money as the Soap Trust has already lost. 1912 G. B. SHAW Let. to Granville Barker 1 May (1956) 182 We went to the aeroplanists’ sheds again... Flying just about to begin as usual. 1928 Amer. Jrnl. Internat. Law 22 81 After interning the Germans resident in Siam she [sc. Siam] sent an expeditionary force to France composed largely of aëroplanists. 1940 Hammond (Indiana) Times 19 Aug. 4/6 The aeroplanists, if they once got their machines above the pollen cloud, would just fly back to the fatherland and report that our country was permanently under a pall of poison gas
Airplane
1. In form air plane. A horizontal plane in the air on which an aircraft, bird, etc., is regarded as flying. Obs.
1868 Eng. Mechanic & Mirror of Sci. 24 Apr. 91/2 We have yet to see the ‘aëroplane’ with buoyancy sufficient to sustain 150 lb., or with apparatus sufficiently light and portable to make headway on an ‘air plane’. 1874 Belgravia 4 168 Not all the rudders and flappers can ever enable the aeronaut to navigate his machine horizontallyto one hand or the other of the air-plane on which he is sailing. 1891 Amer. Naturalist 25 797 Its [sc. a bird’s] weight is resolved into a normal and parallel component by the air plane of pressure beneath it.
2. a. = AEROPLANE n. 1. Now rare.
1896 J. CHALLIS in Invention 13 June 380/2 The combined use of the screw and aeroplane (why not call it air plane) principles. 1907 Washington Post 28 Apr. (Misc. section) ¼ My idea of the airship of the future. Somewhat on the order of a box-kite, with projecting air-planes on either side. 1911 Times 15 Feb. 8/3 The rudders, airplanes, &c. are worked from the gondolas [of the airship], where compasses and other navigating appliances are also arranged.
b. Chiefly N. Amer. = AEROPLANE n. 2. Also attrib.
Airplane became the standard U.S. term (replacing aeroplane) after it was adopted by the National Advisory Committee for Aeronautics in 1916. Although A. Lloyd Jones recommended its adoption by the BBC in 1928, it has until recently been no more than an occasional form in British English.
1906 Sci. Amer. 29 Dec. 487/1 Air-plane is a much better word than aeroplane. It is as good etymologically and much better when it is spoken. 1907 Westm. Gaz. 19 July 4/2 It is this ease of going against the current, with no motive force in evidence, that is the despair of the aeronauts with their air-planes. 1917 N.A.C.A. (U.S.) Rep. Nomencl. Aeronaut. 31 Airplane. This term is commonly used in a more restricted sense to refer to airplanes fitted with landing gear suited to operation from the land. If the landing gear is suited to operation from the water, the term ‘Seaplane’ is used. 1933 E. A. POWELL Slanting Lines of Steel xi. 172 This was no airplane bomb, but a projectile from a super-cannon. 1978 D. B. THURSTON Design for Flying ix. 112 If the airplane is intended for the private market, thin skin could be used and the weight saved converted to useful load. 1995 Mail on Sunday (Nexis) 1 Oct. 32 No one with any sanity would fly in an airplane if they could walk. 2005 Chicago Sun-Times (Nexis) 4 Nov. 26 New rules that would allow travelers to bring items such as sharp knives and scissors onto airplanes, rolling back post-9/11 prohibitions.
Aerodrome
1. A place where a balloon or flying machine is housed; a hangar. rare. Now disused.
1902 Westm. Gaz. 29 Jan. 6/2 He. soared above the Prince’s castle to the aerodrome. 1921 ‘M. CORELLI’ Secret Power viii. 80 The lady of many millions had commanded an air-ship to be built with an aerodrome for its safe keeping and anchorage.
2. Originally: a tract of open ground set aside for aircraft to fly over in flight trials and flying contests (now disused). Later: a large tract of open level ground together with runways and other installations for the operation of aircraft; a small airfield, esp. a private or military one.
1908 Westm. Gaz. 14 Aug. 7/2 An ambulance station is situated immediately outside the Aerodrome, and several members of the ambulance corps were on the spot in the space of a moment or two. 1909 F. LANCHESTER in Flight 2 Jan. 13/1, I regret to see that the misuse of the word ‘aerodrome’ is receiving support in your columns... I suppose because a hippodrome is a big open space for horses, you think that an aerodrome should be a big open space for flying machines. 1911 C. GRAHAME-WHITE & H. HARPER Aeroplane iv. 124 A circle had been whitewashed on the aerodrome to act as a mark in which the aviators were to descend. 1922 Encycl. Brit. XXX. 48/1 The London terminal aerodrome at Croydon, Sur., may be taken as typical of a modern air-port for commercial traffic. 1934 Jrnl. Royal Aeronaut. Soc. 38 725 There was erected the aerodrome beacon. 1939 Flight 19 Oct. 309 Fighter pilots in crews’ quarters on an aerodrome. 1940 LD. ALANBROOKE Diary 15 Aug. in War Diaries (2001) 100 Picked up plane again at Grimsby aerodrome and flew over Humber estuary examining forts and Spurn Head defences. 1953 R. CHISHOLM Cover of Darkness ii. 30 At last I was given a ‘green’, but the dim pattern of aerodrome lights made little sense by this time. 1969 D. WRIGHT Deafness vi. 60 A private aerodrome at Sywell where we spent afternoons watching moustachioed youths learning to fly pocket biplanes. 1994 S. DAWSON Forsytes (1996) II. ii. 355 On his route into town from the aerodrome there had been the chastening prospect of bomb craters and ruined homes to either side.