We seem to be blurring a whole bunch of things here. First off, whilst ICAO is a little vague about the relationship between aerodrome and approach control services, if we look at PANS-ATM para 7.1.1.2 tells us that aerodrome controllers shall maintain a continuous watch on all flight operations on and in the vicinity of an aerodrome as well as vehicles and personnel on the manoeuvring area. Watch shall be maintained by visual observation, augmented in low visibility conditions by radar when available.
If you look at some of the other provision in the document we find that ADC is, fundamentally, a service orientated around VMC operations with radar providing additional information if one has the luxury of having it. Management of traffic operating in IMC - and therefore by definition being separated - is an approach control function. The document makes provision for variations to the basic 'rules' by requiring the detail to be 'as specified in letters of agreement or ATS unit instructions' (presumably on the assumption that these will ensure that the aircraft will be kept an appropriate distance apart if they are not operating by visual reference).
Technically, all of the IFR separations required will be provided by the approach controller or through delegation to the aerodrome controller in accordance with local procedures.
Unless you have a particular interest in UK procedures, don't get confused by some of the comments in this thread relating to ATSOCAS and the very unusual UK system which effectively means that aircraft get a control service when they are outside controlled airspace in some situations.
In practise, an aerodrome controller will manage that 'safe, orderly and expeditious flow of air traffic on and in the vicinity of an aerodrome' by watching where the aircraft are and being happy that they will not collide or by placing the obligation on pilots to ensure that aircraft do not collide by, for example, telling one pilot to follow another aircraft. Of course, passing suitable flight and traffic information makes all of this a lot easier.