A part = union
Apart = separation
A part is often followed by the preposition "of".
Part acts as a noun when combined with the article, A.
Example: He is a part of our lives.
However, the two words do not always appear together. The article (a) can be dropped before (part of) if there is no adjective.
Example: A part of the world lives in poverty.
If an adjective appears before (part of), then the article is used.
Example: A large part of the world lives in poverty.
Apart usually acts as an adverb. Apart is often followed by the preposition "from". When combined apart from is similar in meaning to aside from, or with the exception of.
Apart from being splashed by two crazy guys jumping in the pool and getting totally soaked, I had a good time at the pool party. (true story)
Apart can also act as a phrasal verb when preceded by certain verbs. The verbs can change tense.
Fall apart - After I received the bad news, I fell apart. I was very upset.
Tear apart - The divorce tore the family apart. The love was no longer there.
Tell apart - I was difficult at first, but now I can tell the twins apart. One is good and the other is bad.