A full backup is exactly what the name suggests. It is a full copy of your entire data-set. Although full backups arguably provide the best safeguard, most organizations only use them on a periodic basis because they are much time consuming, and often require a large number of tapes or disks.
Because full backups are much time consuming, incremental backups were announced as a way of decreasing the amount of time that it takes to do a backup. Incremental backup only backup the data that has altered since the previous backup.
For example, assume that you created a full backup on Monday, and used incremental backups for the rest of the week. Tuesday's backup would only hold the data that has changed since Monday. Wednesday's backup would only contain the data that has changed since Tuesday.
The major disadvantage to incremental backups is that they can be time consuming to restore. Looking back to my earlier example, assume that you wanted to restore the backup from Wednesday. To do so, you would have to first restore Monday's full backup. After that, you would have to restore Tuesday's, followed by Wednesday's. If any of the backup happens to be missing or damaged, then you will not be able to perform the full restoration.
A differential backup is similar to an incremental backup in that it starts with a full backup, and subsequent backups only contain data that has changed. The difference is that while an incremental backup only includes the data that has changed since the previous backup, a differential backup holds all of the data that has altered since the last full backup.
Assume for example that you wanted to create a full backup on Monday and differential backups for the rest of the week. Tuesday's backup would contain all of the data that has changed since Monday. It would therefore be identical to an incremental backup at this point. On Wednesday, however, the differential backup would backup any data that had altered since Monday.
The benefit that differential backups have over incremental is shorter restore times. Restoring a differential backup never requires more than two tape sets. Incremental backups on the other hand, may require a great number of tape sets. Of course the tradeoff is that as time progresses, a differential backup tape can grow to contain much more data than an incremental backup tape.