Scopes with Gaussian response have gradual rolloff from very low frequencies, so amplitude accuracy will be impacted. -3dB is 30% percent amplitude error. So for a Gaussian response scope, 1GHz scope will have good amplitude accuracy up to cca 300MHz or about.
Scopes with brickwall response will have very good amplitude accuracy, up to cca 90% of it's stated bandwidth.
Also, scopes with Gaussian response will (by virtue of it's response) slow down any edges that are faster than it can handle (by filtering out all frequency components higher than some frequency).
It will behave same as if you took 1 GHz brickwall response scope and put in 500-600 MHz low pass filter in front of it. Simply as that. It won't show any artefacts (like ringing) because you filtered out parts of input signal that would make problems. Basically, it shows nice pulse from Leo's 30 ps pulser, because it took 30 ps rise time pulse, and filtered it (slowed it down, converted it ) to a pulse with 500ns edge, that is not faster that what is scope's rise time.
And that is exactly what Nico demonstrated (thanks for that, you saved me time of making filter, I had same idea, but not filter at hand).
Basically, whenever you look at the pulse that is seriously faster than than rise time of your scope, you get overshoot. You can fight it by deliberately making analog part of scope slower, oversampling by much larger factor (not very practical or cheap at 1GHz and up), or just accepting the fact that it will overshoot if you try to drive it faster than spec.
Actually, I can argue that after learning of how it all works, I like brick wall response better.
Here is why:
1. Good frequency amplitude accuracy up to 90 % of full bandwidth. I can accurately assess 868 MHz amplitude with a 1 GHz scope.
2. Good rise time measurement accuracy. It will overshoot if you drive it fast, close to the edge of spec , but it will give more accurate rise time measurements right there up to the limits of specification.
3. Overshoot is there only if edge is faster than scope's rise time. Which is actually useful information. It tells me edge is faster than what it can handle, by overshooting..
So overshoot is actually useful. It shows you you are trying to look at the signal that is too fast for your scope, instead of happily hiding anything it doesn't like... You fight it by getting faster scope, not one that hides it better...