Probably answered before but i'll take a stab at it anyways. The most "common" situation you will use this button is in the sim when you've gone around (missed approach), single engine and the missed altitude was captured fast enough before you had time to select another pitch mode (FLCH or VNAV).
i.e lets say your missed approach altitude is 2000 ft (AGL) and it went straight from TO/GA to ALT while you were cleaning up the flaps or going to your 'planned go around flap setting'. In that situation the verification of CON thrust would require you to go and select it in either the THRUST LIM page or using the CLB/CON button as to not bust the 5 minutes T/O Thrust limitation on the engines.
That's basically the situation you will most likely use this button for.
To answer your second question about the diff between using this and the ENG OUT selection on the VNAV page. It's not so much a difference but rather using ENG OUT on VNAV does absolutely nothing if you're not in a VNAV pitch mode. Sometimes (especially in the SIM) if you're going around on a heading and an altitude (great idea to simplify life when you are dealing with non normals and no terrain/obstacles considerations) and you don't want to mess around with VNAV you will select FLCH or just let it capture with ALT as described above. In the former situation, you will find yourself using this button in lieu of selecting ENG OUT on VNAV. If you use FLCH, it will make the transition from G/A to CON for you.