I've been to both sides on a number of occasions so I feel equipped to answer your question:)
It really depends what you're looking for. Southern Cyprus is more of a typical holiday resort destination, and very crowded with British tourists in the high season.
The main resorts are Paphos (port, nightlife, shops, restaurants, historical sights) Ayia Napa and Protaras (very 'Brit abroad', with lovely, albeit crowded, beaches, such as Nissi Beach and Fig Tree Bay).
Limassol and Larnaca are also popular resorts but I can't say I was impressed, the waterparks were great but that was about it.
I loved the capital (Nicosia/Lefkosia), its very relaxed but pretty much dead in the summer when the local Cypriot townies go to the beach for the summer. However, its so laid-back, they have a great mix of international high street (Zara, Topshop, Debenhams, M&S) and designer (Louis Vuitton, Versace). The city Cypriots seem to spend their day shopping and smoking and drinking coffee in posh cafés and then partying in the evening. Zoo is the best nightclub there, but there's lots of clubs and bars to choose from. There's also archaelogical sights, such as the old Viennese wall around the centre, plus a handful of museums.
The green line that divided Lefkosia (UN patrolled) is interesting to see, the old shops and buildings in 'No Man's Land' are literally stuck in time, and haven't been touched since 1974. There's a small exhibition on the South side of the line, at the Ledra crossing point, explaining the events of 1974, I found it very touching, and sad.
You can now cross the line into the North with just your passport, but try to make sure you get your Turkish stamp on separate piece of paper, if you can. You'll notice a difference as soon as you enter the North. Bear in mind its not in the EU, and is less developed than the Turkey itself and the Republic of Cyprus, and English is less widely spoken. You'll find less in the way of tourist informations etc, so you're pretty much on your own, as somebody above pointed out. But that could be a good thing if you're a bit adventurous!
There is little to do in the Turkish side of Lefkosia, but you will be rewarded if you hire a car and venture further North past the mountain range (can't remember the name, sorry) and visit Bellapais (an old monestary, with a few restaurants and shops nearby) and then to Kyrenia/Girne. Here you'll find a beautiful port with plenty of shops with local crafts etc, plus a handful of bars and many restaurants serving fresh fish and seafood.
Overall, I found both the Greek and Turkish Cypriots to be very open and friendly, though it was easier to communicate with the Greeks, as 99% seemed to speak perfect English. It might be beneficial to learn a few Turkish phrases before venturing into the North. I'll be honest, some of the places in Northern Cyprus were very run-down and unattractive, and many Orthodox churches have been either left in sorry state or converted into mosques. But you will still find many places of natural beauty, and I really warmed to the Turkish Cypriots, and no doubt you will too.
Hope I've helped, email me if you want any more info:)