The difference in the above answers is the difference in the terms download limit, and bandwidth limit. I'm almost sure you'll have a bandwidth limit. This means that each and every byte sent to your computer counts towards your limit. I am not sure how big the file sizes are on youtube. They are converted from video into a flash format, this will be smaller, but with audio too you'll find you start nearing your 2gb limit very quickly.
This 2gb limit, is this per month or per day? If it is per month I would be careful about how much youtube you watch. A month of youtube will fly past the limit. If it is a daily limit you can be much more relaxed, the chances are you won't watch 2gb worth per day.