They are not "I get it." or "I got it.". They are only "Get it." and "Got it.". I'm wondering what's the difference between them.
Get it on its own would normally only be used as either an imperative (telling someone to get something) or a question (asking if someone understood; [do you] get it?). I get it (meaning I understand) is not usually shortened to get it.
Got it can be used in a few more ways. As a statement ([I've] got it), it means that the speaker understands, or the speaker physically has something, or (in a similar way to I've got this) that the speaker has the situation under control.
As a question ([have you] got it?) it can be used to ask any of the above: if someone understands (the same as get it?), if they physically have something, or if they have the situation under control (like have you got this?).