What's the functional different between a skillet and a saute pan?

Skillets are not designed to hold much liquid; as you point out, the curved sides are optimized for easy flipping and turning (i.e. with a spatula). They also normally do not come with lids; I'm sure there are some out there, but even my All-Clad skillets didn't.

You can really only use a skillet for high-heat searing or frying due to their low profile. Don't bother trying to poach or even shallow-fry in one.

Sauté pans, on the other hand, have more height, and usually do come with lids. They're a little heavier, and don't make it as easy to turn, but they can also be used for shallow-frying and moist-heat cooking methods like braising or poaching.

I've seen people make sauce or chili in sauté pans as well; sometimes they're more convenient if you're only going to make a small quantity, since the large exposed surface area (relative to a saucepan) makes it easier to reduce sauces. It's a no-brainer if your recipe starts out with sautéing garlic, onions, vegetables, etc., and then has you adding the liquid ingredients; if you have a large enough sauté pan then you don't need to bother transferring to a saucepan afterward.

I think you've got it backwards; of the two options, a sauté pan is definitely the more versatile option. I use both, but if I only had the space or budget for one, I'd choose the sauté pan, because it can do everything a skillet can do (just not quite as well) and many more things a skillet is useless for. A skillet is really only necessary for people who do a lot of pan-searing.

Note that as commenter Owen suggests, there are "French skillets" sold which have higher bases; these are not really skillets, and are not normally sold with lids, but assuming you can find or improvise one, they would make semi-decent multi-taskers. I maintain that cast-iron skillets, while taller, are not nearly as versatile due to their reactive material and weight - I only ever use mine for searing/grilling.