Translation - Italian-Latin - mi manchi tanto amore mio

multum (=multo) --> tantum (=tanto)

Sorry dear, I didn't catch what you meant.

Sorry, I have made a mistake: multo--> molto

And what I meant is:

Mi manchi molto --> Multum mihi dees

but

Mi manchi tanto --> Tantum mihi dees

Am I right?

In Italian, when acting as adverbs, molto e tanto are (nearly) always interchangeable.
Molto is just a little bit more formal. But basically there's no difference in meaning.

Does Latin make a difference between them?

There is not much difference, but it is indeed.

multum = very much (bardzo)
tantum = so much (tak bardzo)

Well, then let's keep that difference. I opt for "tantum".