Достигать is simply "to reach", "to achieve".
It makes you think about the road to the goal, about a process of getting closer and finally reaching it.
"Добиваться" is to achive what you need in struggle.
But it is also is used in combinations like "добиться нужного результата", where it doesn't make you think about stuggle, rather about multiple attempts.
Comaparing to "достигать", добиваться makes you think either about obstacles on your way or about failures. That is достигать is getting closer and "finally" reaching, and добиваться is meeting resictance or failures, but finally obtaining it.