"She is here with me."
Translation:Ona jest tu ze mną.
For pronunciation reason, it would be hard to pronounce "z mną". It happens before some clusters of consonants.
mnie is Genitive=Accusative = locative = sometimes Dative
mną is instrumental
They are perfectly synonymous, 'tutaj' is a bit more emphatic, I guess. I really don't recommend putting any of them at the end of the sentence, but if you really want to do it, at least go with "tutaj".