In the past, tape was the solution. The problem is that this still tends to be the answer. Either backup admins are complacent and don't want to change, or decision makers are unwilling to look at new ways to do backups.
With the ever increasing capacity of disk, it's not a matter of not having the space. The price for disks such as SATA, while not the fastest on the block, is affordable, so it's not a matter of price. Could it be the speeds? I don't see how, especially since disks are faster than tape for spooling data. Disk can also handle more streams (more data can be written at a given time in multiple paths or from multiple sources). In general, tape can handle a single stream of data. (Backup software is changing this; just wait for the time you have to recover from the tape...). Maybe it's just a matter of understanding a disk-based backup solution. In that case, I'll go over the basics.
A standard tape backup process involves running a tape drive or library from a backup server. This server's role is to connect with the other servers on the network and pull their data across the network and place it onto the tape. Simple enough. Doing a backup to disk is just as simple. Most of the latest backup software can handle writing to disk as an alternative to tape. Change the process a little and have the backup server write to disk instead of the tape. The backup software can write faster to the disk since it's continuously moving, unlike tape drives that stop, rewind, and forward to place data onto the tape. Since writing to disk is faster, the backup jobs should complete sooner, meeting the ever shrinking back-up window. An even better benefit is that once the data is on disk, you have the rest of the day to get the data to tape for a second copy or for offsite (long-term) storage.
Disk level backups have another benefit: better restores! The real purpose of backups is not to just have the data, it's the ability to restore the data. For anyone who has had to restore files, this process can be long and painful. By keeping backups on disk (online) restores can happen much faster, since the restores are disk to disk. There's no need to call back tapes or to load and catalog them. A second benefit is that you can actually permit the users the ability to do the restores for you! (This is based on the backup software features.) While these are simple restores, imagine rebuilding a server that crashed. Assuming that tapes and backups ran correctly, this would still involve getting new hardware, reloading the OS, the backup agent, loading tapes, and restoring the data to the new server. With disk backups, there are no tapes to load and the restore is much faster. Should the backup process fail or the data corrupt, you'll know sooner and can adjust accordingly.