There are plenty of English speakers here who know Italian, Elisabetta, so feel free to give us a sentence in Italian to translate.

I come to eat.. vengo per mangiare.

I come eating.. vengo mangiando, cioè vengo e sto mangiando, cioè vengo a mangiare.

there is an explanation for this in articles page "Verb Patterns in English: Gerunds and Infinitives" check it out.

Here are a couple of examples of when each is used:

Do you want to come walking with me?

Are you coming for a walk with me? or Are you coming to dinner tonight?

Give us your Italian example and I will try to translate it for you.

I can't give you an example, because I've just read on my tablet an article about verb patterns: when to use in English to+infinitive or ing form. There were some examples with other verbs, to try, to stop (now I understand their difference), but any example with the verb to come. I can give you an example in Italian language, but I don't know if you speak Italian!