In addition to your standard melee ability from tapping the melee button every class now also has a Heavy Melee attack used by holding down the melee button. The heavy melee activates your omnitool, which crafts a different instant melee weapon depending on your class (they have separate models and animations but the effect is the same for all classes). In addition to doing more damage than the standard melee it also has a useful knock-back/down.
Many more cover related moves, that do seem to be more intuitive. If there's an enemy hiding on the opposite side of a piece of narrow cover, you now have a fairly devastating pop-up and melee them ability (though for this reason enemies tend to fall back or push forward when you dive into cover opposite them so you need to be quick). Movement in and between cover has been improved with "swat turns", diving between adjacent cover areas, turning corners covered, diving out of cover, diving into cover and mantling all improved and far more intuitive to use. Mantling cover while running is very simple with a quick double-tap of the cover button as you approach the cover.
There are differences in attack chaining (worth bearing in mind when you're commanding your squad), you get far more effect by combining like powers than going for different types, so chaining two biotic powers together (or two tech powers) has a far more devastating effect than chaining a biotic power with a tech power. The achievements also encourage this behaviour, with counts for combining two tech or two biotic powers, and no count for combining a tech with a biotic power.
Cerberus now have a range of ATLAS mechs (hulking, big, two legged fighting suits with a rocket launcher and chaingun mounted) you can hijack these if you're careful.