need to know difference in abbreviations and acronyms and
which one is correct and why
a great and prospers new year.
an great and prospers new year.

There is no abbreviation or acronym in your phrases

The correct usage of the article adjectives a/an is this: use "a" when the next word begins with a consonant, and use "an" when the next word starts with a vowel.

a great and prosperous New Year would be the correct phrase.

an optimistic and prosperous New Year would use the an because the next word "optimistic" starts with the vowel o

That all said, it's all about the sound of the first letter (as opposed to whether it's a vowel or consonant).

For example:

An hour [correct]
A house [correct]

Also:

An RTA [correct because R starts with a vowel sound]

Here's a lesson on using an and a with abbreviations and acronyms.