Photoshop seemingly tries to retain glyphs of the text object somehow in proportions when a preset warp is applied. Try the same warp to a rasterized text and you will get the same as in Illustrator.

In Illustrator the only way to get the same for a text object as in Photoshop is to apply Envelope distort > Make with Mesh. Select 1 row x 3 column mesh and edit the nodes manually with the direct selection tool. It's difficult to get exactly the same result.

An example:

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/IKDa5.jpg