"Separated" versus "separate"

Both the sentences are correct.
Separate is an adjective meaning forming or viewed as a unit apart or by itself; separated is the past participle of separate, which means cause to move or be apart.

Most of the people will consider those sentences synonyms.

_{[Reference: the New Oxford American Dictionary.]}

'separate' is an adjective, 'separated' is the past participle of the verb 'to separate'. therefore, 'separate' means the separation is natural and permanent, 'separated' means the action of separating has been performed at some time in the past.

'Keep both of them separate' means, "they are in two different places, don't put them together". 'Keep both of them separated' means, "they have been moved into different places, don't put them back together".

Both are correct.

"Keep both of them separate" implies more "for now". It's not perpetual.

"Keep both of them separated" is more permanent, as in they should always be separated.

I'm sure someone else knows the proper grammatical names (future/present tense terminology), but that's how they differ.

This is just how I would use them differently, but it is subjective; they mean the same thing.

Keep them separate

or

Keep them separately

I would expect that to be talking about objects, that are normally separate and should remain so for a good reason, such as the constituent ingredients for nitroglycerine.

Keep them separated

I would expect that to be talking about people, for example two unruly school children who have been caught fighting.

Both are correct.

Keep both of them separate means move both things to another place.

Keep both of them separated means move each thing away from the other.