In the computer world, they are basically the same. Technically: Upgrade - To improve something. Update - To change to the most recent version. It can be an upgrade, downgrade or removal of features.

Upgrade usually means to make the level higher or make something better than it is. Like it was 1 level character in a game and than became 80 level character. Update is just 'keep up to date', perhaps better, but not much difference or extra changing. You can upgrade your old computer (by changing processor and memory for example) and you can update an application or program up to current (actual) version.