There really is no practical difference; the dictionary definition of a soup is:

a liquid food made by boiling or simmering meat, fish, or vegetables with various added ingredients.

Which also applies to any stew you can conceive of.

The technical, highly-nuanced difference is that of emphasis and intent. Stewing is a method of cooking the solids (specifically, a slow, moist-heat method). When you make a beef stew, you are stewing the beef, which says nothing about what you're stewing it in. On the other hand, when you make a chicken soup (or a chicken stock or broth which is the base of a chicken soup) then your objective is essentially to make chicken-flavoured liquid - to extract the flavour of the solids into the liquid. If some flavourful solids remain, then that is incidental as opposed to intentional.

In practice, some flavour extraction is going to happen with a stew as well, it just so happens that the principal aim is to cook the meat/veggies. A soup is more likely to contain raw or barely-cooked ingredients, and a stew is more likely to preserve the original flavour of the solids and/or liquids, but that's a very broad generalization and what it boils down to (ha ha) in practice seems to be largely dependent on the culture and the dish itself.

A stew is not simply a thick or chunky soup, despite the fact that a lot of people think of it that way. As above, that seems to be more common with stews, but it's not part of the definition, and the French have half a dozen categories for thickened soups that could easily be described as having the overall consistency of a "stew".

Red pizza sauce is often (but not always) two things:

1. Thicker. Thinner sauce will tend to run in the oven and also steam the pizza crust as it cooks - if loaded with toppings, otherwise thin is fine. Depending on the crust, the heat of the oven, the toppings above sauce, and how watery it is, this may not be needed. If you've just got some crushed tomatoes and a few bits of cheese in a super hot oven - the sauce should be fine without reducing beforehand. If you've loaded up a bucket of sauce and a pound of cheese, precook and reduce the sauce.

2. It's often simpler. Many pasta dishes like spaghetti highlight the sauce and hours of simmering happen for the sauce to bring it to perfection. They're all about the sauce. Pizza is really about the crust and what highlights it. Many of the best pizzas are simply topped. Crushed san marzano tomatoes (sometimes) reduced with a hint of salt is often all I do, and franky, it's enough. I'm not talking about the jarred varieties here, but what you might cook at home. Jars in the store, the main difference is just how thick it is.