There are many train companies operating trains and subways in Tokyo. Suica is issued by JR and Pasmo is by other train companies. When you actually use it to pass the gate of trains/buses or use that to buy something at stores, there are no differences. I like the design of suica card better, the penguin is cute :)
Usage-wise there is virtually no difference. The only real difference is that if you choose to refund yourPasmo card, there is no refund charge. Suica and other cards will pickpocket you up to 220 yen refund the balance.