Separate can be a detached object.

Part can be a piece of something, not necessary detached or split.

I think that's the main difference between them two. Separate is an adjective, a part is a noun. 

When separate is a verb its meaning is to detach, to split an object into a few parts, where parts are pieces of the object. Like a wheel of a car, for instance.

I assume you're asking about the verb forms of both words. The verb "separate" has a few uses.  It's often used to describe separating things that are closely attached or connected, either physically or emotionally. You can talk about "separating the meat from the bone" (by cutting or pulling the meat away) or "separating a mother from her child" (by taking the child away from her). If you glue two pieces of wood together, they may be difficult to separate.

To "separate" things can also mean simply moving them apart from each other. If two kids in a classroom always misbehave when they sit near each other, the teacher may want to "separate" them by making one of them sit at the back of the room.  

You can also use the verb "separate" intransitively, to describe things moving away from each other. If you put oil and water into a cup together and mix them, they will naturally separate -- they won't stay mixed, but will move apart. If two people are in a relationship, and then they split up, you can say that they are "separating" (they're leaving each other).

The verb "part" shares most of the same meanings, but it's less commonly used, and can sound a bit more poetic or old-fashioned. 分離 could be translated as either "separate" or "part," but if you're translating poetry or song lyrics, I think "part" will usually sound better.

The most common uses of "part" are intransitive, meaning they don't take an object.  When you "part from someone," you leave them (usually for a long time or permanently, such as when you end a relationship, or move to another city). When you "part with" something, you give it up and don't have it anymore (much like 捨 -- 不捨得 would be "unable to part with").