Full Backup *backs* up includng all the stuff like SQL Related stuff anf Forum DB and other stuff (Ideal if you use anything related to MySQL and PHP..Use this if you have Forums/CMS in your site. If you do not want to loose your Member Database and other stuff)
On the other hand , Home Backup just backups the "/home/youruserid" only , so it copies everything and nicely zip it for you. Ideal for those who wants to backup everything
I suggest you to do both of them , just to be in a safer side ...Backup SQL DB aswell , Backup home folder aswell and make a full backup...Just to be safe...:d
I usually put use FTP to upload it to 3rd party server and keep it there is a safe place..Or you can download staright away..Go fnd a Cheapo Free Hosting that gives you quite amount of space and that allows you to host file. Then FTP it to there...Its safe to keep backups somewhere else aswell