What size are small, medium and large tomatoes

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.

The difference in their name is - not surprisingly - because of their form. Cherry tomatoes are round, while grape tomatoes are more oblong. Wikipedia mentions that grape tomatoes are 'sweet as cherry tomatoes', so not much difference in flavour according to this. Perhaps grape tomatoes are more similar to small plum tomatoes.

This site claims grape tomatoes to be sweeter, having a thicker skin and a lower water content than cherry tomatoes. Interestingly, it also says that grape tomatoes are a cross between cherry tomatoes and other tomatoes.

The claims of the second site are in agreement with this site ("grape tomatoes have thicker skin, lower water content and intense sweetness"). It also states "a grape tomato is half the size of a cherry tomato", so perhaps they are smaller. Due to the lower water content, grape tomatoes have less chance of 'squirting' when being bitten into.