First I warped a text as a wave in Photoshop. Wrote the same text (the same font, size and other properties) in Illustrator and it warped differently - the warping is almost unnoticeabe although the bending is the same (50%) in Photoshop and in Illustrator. Why this happens and is there a way for this text to be done the same in Illustrator? Thank you!
Try using Illustrator's "Flag" warp. It seems to be closer to Photoshop's "Wave" warp.
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:
