1 Although there many advantages to living in the city I prefer live in small town.
2 One of the many reasons America is such a great country is the diversity you
see around.
Question sentence 1 : What is different if I write : Although there many advantages
to live in the city...... what is a different meaning to living and to live?.
(1) Although there are many advantages to living /*to live in the city, I prefer to live in a small town.
Your example is fine where "to" is a preposition followed by the participial living clause, but plural "advantages" cannot normally take a to- infinitival clause as complement, so "advantages to live ..." is ungrammatical.
(2) One of the many reasons America is such a great country is the diversity you see around you.
Generally, every clause requires a verb. The second "is" is the main verb of the sentence and is thus obligatory. Your sentence also contains the underlined subordinate clause, and that too requires a verb, which in this case also happens to be "is".