Most athletic clothes that are called 'compression' really provide very little pressure. Mostly these are really 'form fitting', flexible clothing that touch the skin rather than being lose. There are a number of advantages of compression clothing over lose...
When you are cold, and therefore not sweating, compression clothing will trap the air next to the skin and keep you warmer than lose clothing.
When you are hot, and therefore sweating, compression clothing will spread out the sweat to that it evaporates more efficiently. The evaporation cools the compression clothes, which being next to the skin, helps keep you cool. Very thin compression clothing will keep you cooler than bare skin. With bare skin, the sweat will drip off your body with little cooling effect.
Compression clothing moves far less than lose clothing, reducing chafing. On longer runs, chafing can be a significant source of pain. While lubricants like body glide can help prevent chafing, I believe it is better to have a longer term solution in the form of non-chafing clothing.
The only downside of compression clothing is that it reveals the underlying body shape. For some, this is psychologically uncomfortable. Lose clothing can be worn over the compression layer. While this may be less effective than just the compression layer in warm conditions, it may be better than just the lose layer.
There are clothes that claim greater benefits from compression, such as CW-X. I have used their running tights and not noticed any difference.
Compression socks/calf sleeves generally have much higher levels of compression and are a separate topic.