How more accurate in practice will a STM32 become if we use HSE?
If the accuracy of the internal oscillator is adequate for your application, then by all means use it.  It saves a bit of cost and no board space is used by the parts needed for an external oscillator.
But if you need accurate timing, then a crystal is the way to go.
It's entirely up to you, as the designer, to determine what your application needs.  For some applications +/- 1% is way better than needed.  But for some, like timekeeping for example, it's way too inaccurate.

Is the ADC more accurate?

No, the ADC uses the clock to drive its conversion but its accuracy is not affected.

Is the timers more accurate?

Yes, and this is the biggest improvement.  The internal timers can be no more accurate than the time standard that drives them.  So if you need accurate timing, you will need to use a crystal oscillator.

Does the controller stand against more noise?

No.

Will the ADC be more correct?

I'm not sure how this question differs from the first one.