Hello Alex!
In majority of situations these words are synonyms. Both can be translated as "to finish", "to end", "to terminate", "to conclude", "to end up", "to be over".
Examples:
- Alex acabou o seu trabalho. [Alex finished his work]
- Alex terminou o seu trabalho. [Alex finished his work]
- O filme acabou. [The movie it's over]
- O filme terminou. [The movie it's over]
- A história deles acabou em casamento. [Their history ended up in marriage]
- A história deles terminou em casamento. [Their history ended up in marriage]
However, there are some situations where ACABAR is more specific and is not interchangeable with TERMINAR:
1) When you want to give the idea of "to destroy". Examples:
- Ele acabou com o meu trabalho. [He destroyed my work]
- Ela acabou com o meu brinquedo. [She destroyed my toy]
2) When you want to give the idea of "to give a final touch". Examples:
- Ele acabou o trabalho que eu comecei. [He gave a final touch on the work I started.]
- Ele deu o acabamento na obra. [He gave the final touch on his artistic creation.]
NOTE: you can see that say "Ele acabou meu trabalho" and "Ele acabou com meu trabalho" have two totally different meanings
3) When you want to give a idea of "to consume completely". Examples:
- Em breve a floresta vai acabar. [Soon the forest will be finished]
- O queijo acabou. [The cheese is over]
4) When you want to give an idea of something just hapenned. Examples:
- Ele acabou de chegar. [He just arrived]
- Ele acabou de terminar o trabalho. [He just finished the work]
NOTE: you see that in this context you can even combine ACABAR and FINISH as they are not interchangeable when acabar has this meaning.