The Vatican climate is the same as Rome's; a temperate, Mediterranean climate with mild, rainy winters from September to mid-May and hot, dry summers from May to August.
Most visits to the Vatican City are trouble-free but you should be aware of the global risk of indiscriminate international terrorist attacks, which could be against civilian targets, including places frequented by foreigners.
Vatican City is a city state that came into existence in 1929. It is therefore to be clearly distinguished from the Holy See, which already was in existence for many centuries before that date.