Which one is correct? What is the main differences between them?
I saw him play football Or I saw him playing
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For your particular sentence, the most common sentence by far would be
I saw him playing football
This means that I saw him, and what he was doing at the time, was playing football.
Let's take a different example: perform vs. performing.
I'm a big fan of Marcel Marceau's. I've watched everything of his I can find on Youtube. I even saw him perform once, live, about 40 years ago. I'll never forget it.
I saw my son playing with our little neighbor this afternoon. It looked like they were having a great time.
The key difference is the "once" in Example 1.