לי is literally 'to me'
אותי is 'me', when I am the recipient of an action, not the subject but the object. It is a variation of את , et, which has no translation in English, and shows when something is the object. When I am the object, את becomes אותי.
So for example
אני אוהבת את החתול
אמא אוהבת אותי
You have to find out with each verb whether the word following it takes any particle or not. To love as in the examples above does not, so you see 'et' in the sentence above. But many verbs take a ל or על or מ or ב.
You have to learn for each verb whether the word following them takes one of those prefixes, because sometimes the usage is very different than in English. You just have to memorize it for every verb. Often they are the same as in English, as in the case of 'to love' above. But sometimes not.
For example "to touch" is "לנגוע ב" In English for example you would say "The cat touched me". In Hebrew "החתול נגע בי" Which if you were to translate it literally would be The cat touched on me, but that is not the meaning, it is saying the cat touched me. You would not use אותי or את, because the verb to touch goes with the ב. It is just something you have to learn for each verb.
So when you see sentences which use various verbs try to keep an eye on whether the object of that verb takes any prefix or not.