Android - Must I install Google Apps again after installing a new ROM?

If you don't install the Google Apps package the first time, you won't have access to anything that relies on Google's services framework. This means things like:

• Google Play
• Google Talk
• Google Maps
• The Gmail application

...etc, won't function on your device because they will lack the libraries needed to run.

You do not need to re-install the Google Apps package every time you update CyanogenMod. It will essentially back up your current ones and then re-install them during the flashing process. Simply flash the updated ROM however you normally would.

There's only 2 times you'll want/need to install gapps:

1. On the initial install of CyanogenMod ROM (or pretty much any custom ROM that doesn't have gapps integrated). Since CM nightlies are the ROM only you only need to install it once, Every subsequent install of a nightly only overwrites the ROM contents and doesn't touch gapps.
2. When you know there's an updated gapps package. The gapps package itself is updated from time to time and you'll usually want the most recent version installed.

To be clear, all official CyanogenMod ROMs are the ROM ONLY with all Google apps stripped out (hence having to install gapps seperately). This was due to a cease and desist order from Google to Cyanogen back in 2009 to remove Google apps from its ROMs...CM ROMs have been sans gapps since.