You need to bear in mind that a tilt-shift lens has two different functions, tilt and shift, and they are completely different in their effects.
Most people here have talked about the shift function. That's what allows you to have straight lines in architectural shots, and it's what helps when stitching panoramas.
The tilt function controls the plane of focus. One use of it is for those "fake miniature" shots, which are good for keeping small children amused for 5 minutes but the novelty quickly wears off. But there are two MAJOR uses of this functionality. One is landscape photography, where you can make the plane of focus (nearly) horizontal instead of vertical, so everything in the frame front-to-back is sharp. The other is for table-top applications such as food photography or small product photography, where you can again get the plane of focus nearly horizontal. That allows you to get the top of a cake in focus whilst the table on which it sits is slightly OOF, for example.
Perspective correction can be done in software but the quality takes a hit. Fake miniature effects can be done in software. The other things I've mentioned can't.