The main difference between these commands is in the way they restrict access to their functions.
su (I prefer to think of it as short for "switch user") - does exactly that, it starts another shell instance with privileges of the target user. To ensure you have the rights to do that, it asks you for the password of the target user. So, to become root, you need to know root password. If there are several users on your machine who need to run commands as root, they all need to know root password - note that it'll be the same password. If you need to revoke admin permissions from one of the users, you need to change root password and tell it only to those people who need to keep access - messy.
sudo (hmm... what's the mnemonic? Super-User-DO?) is completely different. It uses a config file (/etc/sudoers) which lists which users have rights to specific actions (run commands as root, etc.) When invoked, it asks for the password of the user who started it - to ensure the person at the terminal is really the same "joe" who's listed in
/etc/sudoers. To revoke admin privileges from a person, you just need to edit the config file (or remove the user from a group which is listed in that config). This results in much cleaner management of privileges.
As a result of this, in Debian-based systems
root user has no password set - i.e. it's not possible to login as root directly.
/etc/sudoers allows to specify some additional options - i.e. user X is only able to run program Y etc.