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.