I'm not sure whether a "the" should be added before the "most" in this phrase. Or are they both OK but have different meaning?
The meaning of those phrases is the same. I've been looking around to see if one is "more correct" than the other. According to the entry for "most" (adverb) in Webster's Third New International Dictionary, Unabridged:
1 a : to the greatest or highest degree : to the greatest extent
"the book that pleased him most"
That would seem to indicate that the correct usage is without "the" in that construction. The definition continues to give an example of the use of "the most" to form the superlative of an adjective or adverb, e.g., "the most beautiful woman," but that's a different usage than in your example.
Both expressions are used as adverb in superlative form. Google shows that the shortened version is preferred, but the longer one is still used all the same.