what can you do with 4E that you can't with 5?
Nothing. You just mod 5e to be exactly like 4e.

Now, what could you do in 4e that you can't in 5e, without re-writing 5e from the ground up to be a completely different game, that's another question. But not worth answering in this context (see the 3e/4e powergamer thread, or the Warlord-discussion ghetto, if you are that interested).
It's the spirit of 5e, not the specifics of it's mechanics and what you might theoretically do with or to them, that make it great.
The main issue I had with 4E was that everyone used magic (how else do you explain a fighter's "Come and get it").
C&GI is a take on a standard-issue genre trope. The gang of minions come at the hero one at a time and he rapidly cuts them down (cf. Inigo Montoya). It was not magic. It explicitly was not magic - like all martial powers, some of which may have been superhuman or extraordinary, but never supernatural.
It was also open to re-skinning, though, so if you wanted to, you could assume it was 'magic' (in some sense) - though it still wouldn't have been affected by Dispel Magic, nor anything that keyed off a magic-associated keyword like Arcane, and still would have been affected by things that keyed off 'Martial.'
That's one of the things about 4e, it was very clear and consistent, and could be re-skinned easily enough, but that very clarity and consistency made modding it unattractive.
5e doesn't have those issues.
5e, OTOH, every class /does/ explicitly use magic. Spells are explicitly magic, and every class but Monk and Barbarian can actually spells (using spell slots), while the Barbarian can use rituals, and the Monk's Ki is also, like spells, explicitly magical.
So your main issue with 4e didn't actually apply to 4e, but does actually apply to 5e. Ironic, that, but not uncommon - the edition war fostered a great many misunderstandings.
I hope you won't hold that against 5e, though, since, as a DM, all you have to do is ban a few-sub-classes (EK, AT, maybe TB, but really, it's just a few nature-oriented rituals) and change the definition of Ki to fix it up to your liking!
You think 2E is better? THAC0 anyone? What makes it "better"?
No, he found 2e closer to what he wanted in some ways. That's just a preference, and only for aspects of the system, not the system as a whole - probably not THAC0, for instance.
How does the game system affect your campaign?
In a lot of ways, but a good DM will leverage them to get what he wants out of the campaign.