What is the difference between a "common" and a "park"?
I've checked some of the dictionaries I have, they say that common is:
1) a large area of open land in a town or village that people walk or play sport on
2) an area of grassy land, usually in or near a village or small town, where the public is allowed to go.
3) (sometimes plural) a tract of open public land, esp one now used as a recreation area
4) a public area or park usually in the center of a town or city
5) is a large patch of grass usually in a village where people used to gather for events in summer. (Ruthi, UK)
6) a public open area in a municipality
7) an often improved and ornamentally planted open space for public use in a built-up area
8) a piece of open land for recreational use in an urban area
9) countable (US also commons) an area of grass which everyone is allowed to use, usually in or near a village

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US:
I never use "common" (lower case) and I don't think I've heard any one use it.

However, there are a few places like Boston Common that use the word.  For all practical purposes, they're indistinguishable from public parks.

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I think I've known more about this word (common) from the defintions that Sasha have cited and from Gary( common with this sense not normally used in American English). Maybe people with different cultural backgrounds have disparate perceptions of the same explanations of a word . For me, a Chinese English learner, the difference between common and park is not very difficult to comprehend: a common ( usually used in Brithish English, according to Gary) is 'an area of grassy land, usually in or near a village or small town, where the public is allowed to go'(Collins), while park is ' (in compounds ) an area of land used for a particular purpose: a business/science park/a wildlife park(Oxford).

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Firstly, as far as I know, a common is term used in Britain only. We certainly don't use it here.
I suspect the difference is a legal, historical, or administrative one - it may depend on what law is used to administer it, and/or if ownership is vested in some governmental agency, and/or how it came into existance historically.