Can you check this text?
 

Hi there!
I wrote a text all by myself, but I'm not sure it is correct. It is a very simple text in which a person named A gives a birthday present to a person named B. Then A find a bag and a watch and asks B if they are theirs (uhm.. I'm not sure this phrase makes sense in English). Can you please check it out? Thanks

A) 誕生日おめでとうございます。これはイタリアのプレゼ ントです。

B) おお、うれしいです。何ですか。

A) フィレンゼのステーコです。
かばんと時計をみました、Bさんのですか。

B) 時計は私のですがかばんはじゃありません。

I'm afraid I'm not really following the conversation at the end. I will just correct the obvious mistakes regardless.

誕生日　＞　お誕生日 (This is because you used the polite ございます)
イタリアの　＞　イタリアからの　(Not incorrect but that it better.)
フィレンゼ　＞　フィレンツェ
をみました　＞　がみえます
かばんはじゃありません　＞　かばんはちがいます

You're right, I made a typo:
かばんと時計をみました、Bさんのですか。==> かばんと時計を見ました、Bさんのですか。

A should say:
I saw a bag and a watch, do they belong to B?
And B is supposed to say: The watch is mine, but the bag is not.

かばんはちがいます: what does ちがいます mean?

It means "Something isn't."

We use ちがう several times per hour.

Ａ：　これＢちゃんの？ Is this yours, B-chan?
Ｂ：　ちがいます。 No, it isn't.

Ａ：　イアタリアの首都って、ナポリ？ Is the capital of Italy Napoli?
Ｂ：　ちがう。ローマだよ。 Nope. It's Rome.

ちがう, is it polite? I mean, can I use it in a formal speech? Or I just use it with friends?

Yes, you can use the verb in any kind of speech, but you would of course say ちがいます when you're using です／ます form.

Uhm.. but then, what is the difference between ちがいます and じゃありません?

There are too many differences to explain.

An oversimplification is just to point out:
ちがいます means "is different"
ではありません (don't say じゃ with ます form despite what teachers tell you) means "is not"

For example:
ア：学生ですよね You're a student, right?
イ：いいえ、違います。 No [that is different from what I am]

You couldn't answer with いいえ、ではありません。 You could say いいえ、学生ではありません though.

Edit: No idea what is up with my fonts, sorry.

So, I can say
これは、貴方のですか。いいえ、これは私のではありま せん。
or
これは、貴方のですか。いいえ、これはちがいます。
but I cannot say
これは、貴方のですか。いいえ、これはではありません 。
Is it right?

Why do you say that? My teacher is a native Japanese speaker and also the textbook uses じゃありません。　Both the teacher and the textbook say that in spoken language じゃありません。is more common than ではありません。

Native Japanese teachers that teach abroad would almost never point out the "less than perfectly natural" phrases in the text. It would only make their job difficult because students like yourself would surely keep asking about them.

How about an observation of a native speaker that lives in Japan, not Italy, which is me? Since I'm not a teacher, I can be more honest about what really sounds natural in our ears.

I don't think I ever say じゃありません myself. I hope not because it doesn't sound good. Where I live, which is mid-town Tokyo, I hear it once in a while though I've never liked it. The phrase has got a terrible balance between the very casual-sounding じゃ and the non-casual ありません. That's just the kind of a phrase that educated people hate to see the most. It lacks the refinedness people expect. I'm pretty sure it would be corrected if a kid used it in a composition in Japan as well. I would surely correct it if my own kid used it.

This may not be the case in Kansai but I doubt you are being taught a dialect.