Which a RAID style system probably won't achieve. HDDs start ahead of tape drives in performance. So it's reasonable to assume that HDD + RAID is also going to perform better than tape drive + RAID.
But it still doesn't make sense to compare HDDs without optimization to tape drives with optimization. Data centers are going to use every affordable optimization at their disposal, so if we want to know "will tape drives be competitive against HDDs" we can't handicap HDDs by comparing mirrored arrays of tape drives to straight, un-optimized HDDs.
With that said, I outlined in another comment[1] why a mirrored array of tapes would have pretty huge drawbacks regardless. Even if you have an expensive array of tape drives (as opposed to one drive with many tapes), a mirrored array will drag your write seek performance toward a constant worst case as the array grows. The same is true of RAID 1 for HDDs, of course, but the worst case seek time of an HDD is orders of magnitude less than that of a tape drive, so HDDs still win.
In any case, if you look at the details of the situation, the answer you come to is pretty boring: tape drives can compete with HDDs in a small slice of real world cases (often when used in conjunction with HDDs), and higher density tape drives will slightly widen that slice.
If you have a situation where you are writing a ton of data, but almost never reading or erasing, then high density tapes will play well to that situation (sans mirroring).
If you have a situation where you are writing a relatively small amount of data over a long period of time, reading it back comparatively frequently, and never erasing anything so that your total storage needs become very large, then maybe a mirrored array of tape drives would make sense, if it were behind a sizable HDD array. But it's a stretch.
