There is a difference in input timing between HF and ST. Specifically ST is harder. Inputs got much easier after that in the Alpha, Vs, and III series.
shoryuken.com/forums is probably a much better place to ask this question.
There are several reasons why playing on an emulator feels different from the actual hardware.
Emulators can add input lag to your controls, keyboards and keyboard encoders can ghost, usb polling rates my affect your control response, emulators may emulate at a different speed than the original, you may be playing on a different speed, different roms have different speeds, you may be playing on inferior quality buttons and joysticks, your computer may change the speed, etc.
Your Xbox 360 has a much easer time of it. It has been coded to run one game on one set of hardware with one type of controls. But even then, I know people claim they can tell a difference even on it.