I have just recently begun using a CPAP machine and intend to take it to France and Italy in two weeks. If I take this in its carrycase, will that use up my carryon allowance or is a medical device a separate issue?

a CPAP won't count against your carry-on allowance.

Usually it doesn't count against your carryon allotment, but best to phone your airline(s).

I wonder if the same will apply when I fly Vueling from Rome to Paris? I know the budget airlines are strict on luggage matters.

crepes - it is not supposed to because it is a medical device and they must stay with you, not be checked. I have traveled with mine a few times and still have my carry-on and a small purse. But as mentioned above, you might want to check with the specific airline.

I should warn you that I get stopped at security each time so they can search the bag that the CPAP is in, that doesn't mean it will happen to you but IME, it has each time I've gone through security.