Grammatically, they are written correctly. However, each has a different meaning.
The first one is Past Tense. That means the action of the person seeing you started and ended before now, which is some specific time in the past. The adverb recently tells to what extent the person saw you. The adverb is modifying the verb saw to mean "not long ago."
The other sentence uses the Present Perfect Tense, which can mean two things, an action that started and ended at no definite time in the past OR an action that started in the past and continues up to now. I'm sure for your sentence, it means an action that started and ended at no definite time in the past.
You see, the Past Tense has a definite time in the past. For example, the person saw you at exactly 2 o'clock yesterday afternoon; with the other, there is no definite time. You may wonder why is that important about what time in the past?
Speaker: The police are looking for Jim. I think he's in trouble.
Alex: Wow! I saw him yesterday at the gym. [a specific time]
Carol: I have seen him at the library, but I forget when. [no specific time can be determined]