It has no disadvantages to you to use the international phone format when calling from a GSM or newer technology cell phone. If you are calling a national subscriber with the international prefix, your call will still be billed as a national call. YMMV in other, older cell networks, if you should happen to find something like that.
The obvious advantage to always use the international format is that you can use the stored number even if you are abroad.
The procedure on how to dial a national phone number from abroad is different from country to country. Although there are exceptions, there are however a few general rules, which mostly work:
If the country you are calling to has no area codes, the national number usually follows the country code without modifications. Using the national, Norwegian number 22012345 as an example, you would dial this number as +4722012345 from abroad. In countries without area codes, it is not uncommon that regular phone numbers start with a 0, where the 0 is a part of the phone number and must be dialled from abroad, e.g. the Italian national number 0612345678 should be dialled +390612345678 from abroad.
If the country you are calling to has area codes, the national number usually follows the country code after omitting the trunk prefix. The trunk prefix is usually the first digit of the area code and is from a technical view not considered as part of the phone number, but as a signal to call someone outside your own area code. In most countries with area codes, like e.g. in the UK, the trunk prefix is 0. So if you have a UK national number 01234 556677, you would dial this from abroad as +441234556677 (leaving out the 0 between +44 and 1234).
As a side note, be aware that the leading 1- in US phone numbers is not a trunk prefix in the same sense that a trunk prefix is used in most other countries (leading digit of the area code), but a prefix usually required if you are calling someone outside your free local call area and hence rather a kind of acknowledgement that you accept the charges for the call you are making. The equivalent to a trunk prefix in the North American numbering plan was traditionally the second digit of the area code, which used to be 0 or 1, while local phone numbers had only numbers 2-9 as the second digit.
One exception from these rules is e.g. Switzerland. Although originally a trunk prefix, all regular, Swiss phone numbers now have a leading 0 as the first digit. This 0 is part of the phone number and must always be dialled within Switzerland, it should however be omitted, when calling a Swiss number from abroad. The national number 0441234567 should be dialled as +41441234567.