Aside from spacing, is there any disadvantage to using groups instead of a Fleet.
As far as I know no. I don't even think the spacing is really a disadvantage.
When should I use a group(they seem much easier to assign and manage) as opposed to fleets which are a bit awkward to manage.
I never use fleets. Ships default unit AI is almost always better than the fleet AI. Keeping my formation pretty is not really what I want them to do in battle, especially when its at the expense of shooting things.
Does the AI perform better in fleets?
IMO definitely not.
Do capital ship abilities which affect the fleet work in groups as well?
Fleet is simply a management construct. It has no significance on abilities or any other game elements. Only how you manage your fleets.
So to sum it up, my advice to anyone who cares about managing their fleet in combat (as opposed to just watching the battles) is to never use fleets.