From what I understand, they both can mean "and" and "also" or something, but I don't entirely get the difference. Please try to explain in as simple of terms as possible!
The first point to note is that the particles も and と have many different uses other than the ones you mentioned, so don't be surprised if you see them used in ways that don't fit this discussion.
When と is used to mean 'and' it denotes exhaustive listing. So if you say:
It means that there is a dog and a cat and nothing else. In contrast if you replace と with や it would mean that there is a dog and a cat and maybe some other animals that I haven't mentioned.
I can't think of any case in which と could be used to mean 'also'.
も can mean 'also'. For example:
I like cats too. (I, in addition to someone else, like cats)
I like cats too. (I like cats in addition to other animals)
も can mean 'and' if it appears more than once:
There are both cats and dogs.
Which is equivalent to "There are dogs and there are also cats", so it still has a meaning of 'also' as well as 'and' in this context.