Depending mostly on the file format, compressing data can save space. As such, not all files can be compressed efficiently with the same ratio.
For example, a 1 MB PDF file will take the same disk space as a 1 MB text file, but the text file will provide better compression results because, unlike text, PDF already is a compressed file format. When compressing files, you can only guess the final space savings; you only know for sure how much space you save when the compression is complete.
Upgrading storage is the quickest route to gain space because packing and unpacking hundreds of gigabytes of data is just not worth your time in the long run. You'll lose countless hours in the process. Save your money, buy one of those terabyte hard drives, and be done with it for years (or maybe months, if you're constantly topping your monthly bandwidth and recording HD programming, for example).
Advantage: upgrading storage.