Hi :) While they can be similar, existence has a few different meanings. For example, it can mean continued survival - like of a certain species of animal (e.g. the tigers' existence was continuously threatened despite ...). In that case, the word presence would not be synonymous with existence. In another context, existence can mean the state of living (like for an organisation having existed for a certain period of time). Existence can also be seen as a way of living and is often used when you're talking about having an 'existential crisis'.
The difference between the two is that when you talk about a 'presence' you can be talking about something existing but not necessarily being seen (i.e. a ghost). Presence can also refer to the physical influence of a group of people in an area <-- that context doesn't work with existence. However, in that context, you could use presence to talk about how the u.s. maintained a presence in a certain place during ww2 or something :)