They're both correct, but they're not variants. The have completely separate meanings. "He walks slowly" means that this person generally walks slowly ie. whenever he walks, he does it slowly. "He is walking slowly" means that he is walking slowly at the time of speaking ie. he is walking slowly right now, even though he may not do so normally.

They are both correct, but they have slightly different meanings. He's walking slowly. (Right now, you see him walking slowly.) He walks slowly. (In walks slowly in general, but not necessarily now.) For example, if you are talking about a friend who is not here right now: My friend hurt his knee. He walks slowly. (good) My friend hurt his knee. He's walking slowly. (wrong, because the friend is not here right now)