It's not really a fair comparison, because the noise bandwidth is completely different between the old 20 MHz scope and your newer 200 MHz scope. What do you get if you engage the 20 MHz bandwidth filter on the Rigol?
If the two scopes have a similar noise figure in terms of V/sqrt(Hz), then a scope with 10x the bandwidth would be expected to show a line more than 3x thicker with no input signal for this reason alone. It's one reason scopes include bandwidth limiting filters that you can turn on. At a guess, when you select GND coupling on your Rigol, it's either a software feature that ignores the ADC output entirely, or it couples the ADC input directly to a fixed voltage and completely bypasses the front-end amplifiers.
Also, one of the major advantages of a digital scope over an analogue one works against it here. On the analogue scope, you see a line whose cross-section accurately reflects the temporal distribution in voltage, ie. it's brightest in the middle and rapidly decays to an invisibly low level as you look above and below. In other words, it has very high dynamic range in the 'Z axis'.
The digital scope's display system, however, tends to exaggerate the brightness of infrequent captures, and though this is normally a benefit as it makes infrequent events more clearly visible, it does make the baseline appear thicker than arguably it should be.