It really depends on where in Italy you are going. However a week won't make much of a difference.
August is the peak time for summer vacation spots at the seaside or the mountains or the lakes. July isn't that far behind. However in August the chances of finding "no vacancy" ("completo") signs is slightly higher than July at those types of locations.
The big cities (Florence, Rome, Milan, Naples) have more vacancies in August, because...well because everybody is at the beach. For example between April and October, the number of occupied hotel rooms in Florence is lowest in August for example.
Venice is packed all year around (not sure if you are going there).
In terms of climate, July is the hottest month, not August. However August is the second hottest month, but at the start of the month there is no difference. Basically between mid July and mid August it is the absolute hottest time of the year in Italy, with temperatures easily in the 100s F (high 30s to low 40s centigrade) in many places (coast and mountains at high altitudes are cooler, that's why everybody goes there).
Not much will be closed. Maybe some shops in the big cities, but not really in the city centers where tourists go.