A differential backup will backup all files that have changed since the last full backup. In other words, if a full backup was done on Monday, Tuesday's differential will backup all changed files since Monday's full. Wednesday's differential will backup all changed files since Monday's full including the files that have changed on Tuesday.
The big advantage to this method comes when performing a complete restore since only the full and latest differential backups need to be restored. The downside is that the size of the differential backup will grow throughout the week and become progressively larger until the next full. This can adversely affect backup windows.
An incremental backup will backup all files that have changed since the last backup, regardless whether it was a full or incremental backup. In other words, if a full backup was done on Monday, Tuesday's incremental will backup all changed files since Monday's backup. However, Wednesday's incremental will only backup files that have changed since Tuesday's incremental backup.
The main advantage to this method is that a lot fewer files are backed up daily between full backups allowing for shorter backup windows. The disadvantage is that when performing a complete restore, the latest full and all subsequent incremental backups must be restored, which can take significantly longer.
The simplified technical explanation could be summarized by stating that both full and incremental backups reset the archive bit on a file (indicating it has been backed up) and a differential backup does not.
To complicate matters however, some software vendors felt this terminology was not adequate and decided to take a different approach. To some, any backup that does not arbitrarily backup all files (like a full backup) is considered an incremental backup. There are then cumulative incremental backups (called differential backups by most people) and there are differential incremental backups (simply called incremental by most people).
Unfortunately, this introduces a contradictory use of the term differential, which is at the source of many discussions about what the proper terminology should be. That said, regardless of which terminology you prefer, what is important to understand is that a full backup will cover all files and subsequent backups between full backups can either reference the last full or the last partial backup.
When attempting to determine which of these technologies is best suited for a specific environment, the differences described above become secondary. With today's high performance hardware, data explosion and compliance issues, the focus has shifted to deciding between tape and disk backup or data replication rather than between full, differential and incremental backups.
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