I'm from Italy and I've moved to Austria. Which are the pro and cons of registering my residence abroad?
I don't think you can generally chose to be a resident or not while living somewhere else. Whether you are considered a resident and for what purpose depends on local law. Some countries (e.g. France, UK) will simply consider you a resident based on how long you stayed somewhere. Others (e.g. Germany, the Netherlands) have mandatory registration systems but it's not up to you to (de)register or not when you move (you might occasionally get away with staying registered somewhere but that's not quite the same). Also, the country you left will not automatically learn about your new registration so if there is some advantage to staying registered, you might also be able to get away with that but it's probably illegal.
In Austria, registering within four months of moving is indeed mandatory, see official government information on the topic.
That said, beside avoiding trouble and doing things properly, there is in fact an advantage to making sure your residence in another EU country is documented and recognized. Under EU rules, after five years in the country, you enjoy extensive rights in the host country even beyond those enjoyed by other EU citizens. In some countries, it might also open a path to citizenship for you or your children, if that's something you are interested in.
While living somewhere and not being registered usually isn't a legal option, you may want to consider differentiating between a primary and a secondary residence when you have two households.
For example, if you work and generally live in Austria, but you still have a house or apartment in Italy, you might want to register your home in Austria as primary residence and your home in Italy as secondary.
I checked not long ago for the situation within Germany, and declaring secondary residences can allow you to deduct various costs from your taxes in relation to separation of work and family in two different households. It is also possible in Germany to register a vacation home for tax-benefits. I can imagine that there might also be tax benefits of these kinds internationally. But tax law is complicated and depends on lots of individual factors and being able to supply proof of fulfilling certain conditions. You have to look at your situation specifically and read through the legal statutes to see what possibilities are available - or hire a tax-lawyer if you have enough money at stake to make it pay off.