We must remember a DSO is NOT a scope in the sense of the older analog scopes like the ones Tek provided, these portray signals in a more "realistic" sense because they are "almost" real time, "almost" in the perspective of just slightly delayed in time to the signals occurring at the probe tip which is shown as a electron beam sweeping across a screen (not considering triggering and retracing). Note the signal of interest is delayed, scaled/amplified and filtered but NOT sampled as in a DSO, and some non-linear effects may be introduced along the way to the screen for viewing. Whereas a DSO is actually a data acquisition system, disguised as a scope which captures data with Analog to Digital Conversion with sampling techniques (both of which are non-linear functions). The sampled waveform data stored (just digital bits now) in memory and displays a rendering of the data (digital bits) captured in memory with manipulations on a computer type screen (pixels) to render what the DSO "thinks" was the original waveform in a from the user prefers. Totally different than the phosphor based screen with sweeping electron beams used with analog scopes. Here, the DSO "thinks" involves all sorts of digital data manipulation, some user requested, some not, and varies highly with DSO brand, settings, and development date.
If one thinks for minute an ideal squarewave has infinite harmonics, which the analog scope low pass filters and displays as the pulse step response on each edge, thus some of the harmonics are attenuated, and higher one towards obscurity. We often think of this "response" as the Step response of the filter, but can also be viewed as a consequence of affecting the harmonics in the original waveform step. However because these harmonics are filtered in real time by the analog filters, and displayed by the pseudo-real time analog screen, the resultant display possess no magic non-causal effects (Gibbs ears).
A DSO behaves quite differently as mentioned being a sampled data system, and displaying results long after the fact and highly manipulated. Especially true wrt to the actual display, and how it's rendered, which involves "waveform reconstruction" in the sense of the discrete time digital data is arranged/manipulated to display an apparent continuous waveform to the user by means display "filters". One must remember the DSO screen is NOT continuous in time nor amplitude, but yet another discrete X and Y system, and the sampled signal digital data which is discrete in time (X) and amplitude (Y) must be "fit" into the display area for the user to interpret!!
Regarding the Gibbs ears, this is a manifestation of the digital rendering of the captured analog signal by the non-linear processes of Analog to Digital Conversion and Discrete Time Sampling, with the effects of both analog and digital filtering involved, and manipulation for displaying on the DSO screen to add further confusion.
So it should be no surprise there are significant differences between the two methods of time domain waveform displaying. One is entirely continuous in time and amplitude (analog), the other is highly discontinuous in time and amplitude (DSO).
Best,