It is very interesting how quickly things change in our industry -- My answer to this question a year ago was highlighting replication or mirroring for highly critical data, disk-to-disk backups for middle tier data and finally, low cost tape backups for the tier 3 backups and archives. The main driver behind this rationale was the cost of tape backup media which was still the cheapest media. At the same time, a newer technology was...
surfacing and slowly getting backup administrators' attention: data deduplication for disk-based backups.
When we look at the major and rapid advances in the field of deduplication (companies like Data Domain are now on the fourth generation of their product), tape has lost a lot of price advantage. With data reduction ratios averaging 20:1 in some cases, deduplication technology has reduced the cost of backing up to low cost disk by the same order of magnitude -- making it that much more competitive when compared to tape.
Add to that the risks inherent to tape media handling such as damage or even worse, media loss. The latter has been plaguing many companies in the financial sector, making the replication component of many deduplication technologies a clear advantage.
That said, tape is still not dead. Data deduplication is not as effective with image files such as scans, pictures, MRI, etc. Furthermore, some organizations have requirements for data encryption at the source, which seriously reduces or even nullifies the benefits of backup data deduplication. Also, with the recent announcement of encryption at the tape drive level (LTO-4), tape technology may have bought itself a few more years.
Check out Pierre's answer to this question from last year.
Go to the beginning of the Disaster Recovery FAQ Guide.