benmeroff

It also bears mentioning that you only need root privileges (and therefore your password) to start servers like Apache and MySQL because they run on privileged ports (<1024) and/or under special, locked-down accounts.
 
If you're not using your computer as an actual "production" server, you don't need these features. For local development, it's much easier to run them from your normal user account (run Apache on port 8080, for example). That way, you also don't have to mess about with sudo and root permissions to edit configuration files.

Terminal displaying your password isn't the only downside: it's also sitting around in plaintext. Not a good idea.
 
The AppleScript method is way better, imo. OS X pops up its standard "Enter your password" dialog, i.e. it uses a GUI, not a shell (that also displays your password).
 
If you don't want to be asked for your password, you should edit the sudoers file. It's not that complicated, and your password isn't sitting around in plaintext. You can also restrict it to specific commands/scripts/programs:
username ALL=(ALL:ALL) NOPASSWD: /path/to/command (Obviously, replace username with your username and /path/to/command with, well, the path to the script/program)