These are so-called light verbs.
Certainly I agree with James K that "take a seat" is more polite than "sit".
But there's more to it.
Light verbs often may be paraphrased by a full verb. But the sense is slightly different.
- to nap vs. to have a nap
- to claim vs. to make a claim
- to look vs. to have a look
- to consider vs. to give consideration
and thousands more.
I think the sense is "to [do something] a little bit and a single time".
For example, someone claims to be Elvis Presley. That means he claims it all the time, or that he regularly claims that. But if someone makes the claim that he's Elvis Presley, it means that he says so maybe once, perhaps as part of a prank or game, but does not actually live his life in pretence.
Or, after a car crash, I could make an insurance claim. That's a single event, so a light verb fits. But if I were on welfare, I would claim benefits, which is regular and ongoing. In this case a light verb wouldn't really fit.
In the same way, "take a look" would mean "have a look once" and maybe "look briefly".
- Could you have a look at this homework? one time action, and gives the impression that the task will not be too arduous
- Could you look at this homework? could give the impression that there is something serious to look at.
It may be worth noting that light verbs are telic, which means they don't go well together with "used to" or "for x duration" and some other constructions