Are there any differences between "update" and "upgrade"?

Update is a shortening of "bring up to date" and upgrade means to put at a higher grade (as in to make better). You can upgrade your your CPU from 2.4 GHz to 3.2 GHz, but your software update can have bugs that break your computer.

To update means to bring someone or something up to date, whereas to upgrade means to raise or improve something to a higher standard. The difference between these two is particularly apparent in the world of computers: an update is not always and improvement!

To update means to make something more modern or up to date; it also means to give to somebody the latest (or up to date) information.
To upgrade means to take something to a higher standard/level; it also means to raise somebody to a higher grade or rank.

Looking in the Corpus of Contemporary American for phrases containing update or upgrade, I get the following data:

                    Instances
computer update     1
computer upgrade(s) 12
hardware update     2 
hardware upgrade(s) 11
software update(s)  30
software upgrade(s) 31

The sentences containing hardware update are both referring to a software tool.

The Hardware Update Wizard will search for an updated driver on your local drives as well as on Microsoft's Windows Update site, and it will install the driver if it finds one.

Select the option in the Hardware Update Wizard that lets you choose the location of the search and select the driver (the wording varies).

From most answers:

To update means to make something more modern or up to date; it also means to give to somebody the latest (or up to date) information.

Well, the most "up-to-date" is not synonym of "modern".

One can have the most up-to-date SQL Server 2005 with all latest updates and hotfixes but it is behind, in being modern, after SQL Server 2008 (even not updated).

For example Service Pack 4 for SQL Server 4 was released on December 17, 2010, i.e. three years after release of more modern SQL Server 2008.

All the above contributions are laudable and basically linked. Hope this scenario simplifies and clarifies the basics.

Updating as explained earlier implies keeping something up to date thus if the functionality of a particular program is to rectify errors from level 1 to say level 3, updating the program implies maintaining or igniting its efficiency to work effectively within the specified levels (level 1to 3).

But to upgrade the program implies increasing its capacity and efficiency to enable it exceed or work beyond the previous lower levels i.e. levels 4, 5 and upwards.

For example think that there are 10 levels and you are at level 1.

If you update a file it will reach directly to level 10 from level1(up to date) If you upgrade it from level 1 you can reach only level 2 (i.e. step by step )