Check out this place!
Check this place out!
Are the two sentences above both correct? and if so, do they have the exact same meaning?
Yes, they are both grammatical, and as far as I know they have the same meaning.
I think "Check this place out" is more common for prosodic reasons: it puts a strongly stressed word at the end. "Check out this place" puts emphasis on "this", and suggests "as opposed to [some other] place".
Both are valid, and reasonably natural, if rather informal. The meanings are essentially the same. The first, with "place" at the end, may emphasize the place a bit more than the other. The second, ending with "out", is more strict;ly informal in usage.
a Google Ngram shows "check out this place" to be more than twice as frequent as "check this place out" in their collection of texts. However, in the "British" sub-collection, only the phrase "check this place out" occurs at all.