I can't find an official definition, but it seems there is no difference, depending on your definition of "lock-down". "Red Zone" seems to be a synonym for "lock-down:"
As week three approached, all of Lombardy was declared a "red zone" and we were ordered to stay in our homes and only leave to buy essentials like groceries and medicine. The next day, the restrictions were extended throughout the entire country of Italy.
As to what "Red Zone" entails exactly, here is the best description I could find (emphasis mine)
Since Feb. 23, the town, along with nine neighboring municipalities in the same corner of Lombardy and one in the region of Veneto, has been designated as a “red zone”: No one can enter, and the 47,000 people within the zone cannot leave.The local Radio Codogno, now known as Radio Red Zone, has been broadcasting news and updates but has also shared airtime with priests to celebrate the 10 a.m. Mass and has organized “storytime” for younger listeners whose schools have been closed. Teachers have started sending out homework via WhatsApp. People leave care packages for friends and family on the perimeter of the red zone. And the municipality of Codogno itself has launched an initiative to collect photos, videos, short stories—and, of course, memes—to document how locals are dealing with these unique circumstances.
As to how many Red Zones there are, the entire country seems to now be declared a "Red Zone:"
Prime Minister Giuseppe Conte on Monday declared the entire country a "red zone," meaning people should stay home except for work and emergencies.
A "red zone" lockdown, implemented this weekend by the Italian government on 16 million people in the country's northern areas including Lombardy and 14 other states, has been expanded to the entire country. Italian Prime Minister Giuseppe Conte on Monday told reporters that "there won't just be a red zone... there will be Italy."