There are several small differences between the Xbox 360 and the Xbox Series X gamepad. No single point by itself would be a very big difference, but overall it sums up. I have both gamepads in front of me, and will try to make a comparison:

• The material of the Xbox 360 gamepad feels “better”. I can’t exactly say why, but I think it’s because of its smooth material on the bottom.
• The Xbox 360 gamepad has bigger analogue sticks, with stronger springs.
• Similarly, the triggers of the Xbox Series X gamepad are “weaker” than of the Xbox 360 gamepad.
• I would have sworn that the Xbox Series X controller is a lot lighter too, but turns out, after weighing them both, that the Xbox 360 controller is slightly lighter. It does not feel this way though, with the Xbox 360 gamepad feeling way sturdier and heavier (but, as said, it’s actually lighter?!?).
• The buttons on the Xbox 360 gamepad feel a lot smoother. They don’t make a “cheap, broken device” noise when being pressed.
  • This also applies to the D-Pad.

I think the last point - the feeling when using the buttons and especially the D-Pad - is the most important one for me. On the Xbox 360 gamepad the buttons feel like actual buttons. On the Xbox Series X gamepad they sound and feel like a fidget toy. Using the D-Pad on the Xbox Series X gamepad is really annoying, because of the noise it makes.