Sauce – Spaghetti sauce consistency / coating noodles

I can give you the Italian answer - first of all, normally we don't use tomato paste to make sauce, but rather to add a tomato "kick" to recipes. Tomato paste is simply tomato puree that has been cooked down to a high degree of concentration.

A basic tomato sauce is made by

1. making a soffritto with onion, carrot and celery (plus other flavors)
2. adding tomato puree, or "pelati", more rarely fresh tomato
3. cooking the sauce down until the taste and thickness is what you want

If you want to use paste instead of puree, the third step has to be omitted or greatly reduced in duration. Keep in mind that tomato paste has its own taste, and that taste will remain in the final sauce.

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.