Why use chopped tomatoes and tomato paste/puree?

Tomato puree is concentrated tomato. So where it is called for in conjunction with other tomato products, be they fresh or tinned, it is an attempt to increase the tomato flavour/presence in the dish without increasing the bulk or water inherent in less concentrated products.

Additionally the flavour of puree, having gone through a reduction process is somewhat different, more intense and rich than straight tomatoes.

Indeed, canned tomatoes tend to be surrounded with juice that is more reduced than would occur naturally. Which is part of why they are used against fresh tomatoes.

Now knowing that you're referring to what is known in America as tomato paste and not actually crushed/puréed tomatoes, the reason is very simple: It's a thickener.

Tomato paste (or purée or concentrate in the UK) has been reduced to remove almost all of the water. When you introduce it into a sauce, it will soak up all of the excess water and make the end result far less watery.

It's not a "thickener" in the same sense as starch; there's no chemical reaction taking place. But when added to a sauce it will give you a creamier texture at the end.

Many homemade tomato sauces I've tried that other people have made are extremely watery and some people seem to prefer this (I know a few Italians who insist that it is more "authentic") and other people claim that using tomato paste is "cheating" somehow. I've never understood that; good cooking uses whatever ingredients are available to produce the highest-quality food. If you're not using crushed tomatoes as a base, then it's often wise to add some tomato paste so that you don't have a big pool of water collecting at the bottom of your pasta as you eat it.