The answer to you question is MAYBE. it depends how much data you have to backup and what kind of data it is. for example, video files and mp3s are already compressed so you would not be able to compress them much further. however you have a lot of documents like spreadsheets, word processor, database files, etc. they will compress much better.
Also you should check out Clonezilla it would make it easy to move all your data from your old drive to the new drive and have you back up and running in minutes.