Spartan II's were kidnapped as children and indoctrinated into being loyal soldiers. They were given all sorts of drug, gene, implant and chemical treatment that made them super strong, fast, smart, resilient, etc. They were created to act as elite operatives in dismantling the rebel groups that were appearing due to the UNSC's inability to effectively govern such a huge number of colonies. Essentially, super-assassins.
When the Covenant attacked they were given a more direct combat role, but still focused on assassinating leaders, retrieving artifacts, and blowing up ships. They were given the powered armour that we all love, which made them even more awesome.
Over time it was realised that the Spartan II's were dying off too quickly, and the cost was too much to make more of them. Thus, Spartan III's were made to be cheap and expendable supersoldiers.
Spartan III's were made from adults from destroyed colonies, given a quick upgrade, and given equal or worse armour. Overall they were worse in every way, but much cheaper and easier to make. Noble Team was unusual in that they had shielded armour, though that could have been because they were on Reach and had easier access to high technology. Jorge was actually a Spartan II, though.