Up time is the amount of time that your phone is powered on.
Awake SHOULD be the amount of time that you have the phone unlocked and are using it. The awake time should stop going up the second you hit the power button to turn the screen off.
Try this: Look at your awake time and write down the time it says right before you turn off your screen. Now wait some amount of time, say 5 minutes. Now check your awake time again. There should be about a 5 minute difference in the time you wrote down and the awake time that your phone is displaying when you turn the screen back on. If there is no difference you have an app or process or something that is still running and you need to figure out what it is because that is eating your battery.
You could also try doing a power cycle on your phone, this will reset your up and awake time as well as restart all running programs which could help your issue. After a while of using your phone, and not using it, you should notice a difference in these two times. If your up and awake are the same then something is running when the screen is off. Do this while unplugged from your charger.
I hope that makes sense.