The portion of Amadeus to which you refer is unfortunately a rather accurate depiction of a practice that has thankfully passed, that of using pounding large staff on stage to keep time. Jean-Baptiste Lully was literally an unfortunate casualty of this practice.
As for Rubato, the Harvard Dictionary of Music offers two related definitions. The main distinguishing factor is that an older sense of the term occurs at a micro-level, changing note durations in the melody without allowing the beat to move around, and that includes accompaniment that closely follows the beat, say in the singing of an aria. This sense of the word carries forward to every popular recording today. You could claim that every commercially popular singer makes use of this sense of rubato no matter how quantized the accompanying tracks are. In truth, here, you aren't robbing so much as constantly borrowing time, which you are also constantly repaying. By this standard, you would have to put Frank Sinatra at the very top of the list!
The second sense of rubato is the "modern" sense in which changes in tempo occur "in all parts at the same time without any compensation". We typically point to Liszt for this sense of the term. However, as a matter performance, the documentary records for this sense of the term go all the way back to Frescobaldi.
Important to your question of the difference in the sense of rubato in the case of Mozart v. Beethoven, my answer would be not too terribly much. Let me explain.
There is a spectrum of time manipulation between absolute metronomic rigidity and absolute improvisatory fluidity approaching chaos. Within that larger spectrum of what is possible there is a certain range of acceptability along the spectrum for any given passage of any given work. By means of physical comparison, a passage is kind of like chop sticks. For any given pair of chopsticks, there is a certain amount of bending possible before you snap and break them. You could make the same analogy with glass. So that range of flexibility can be generalized to given periods, or better, given composers. Bartok, for instance, gives us not only metronome markings, but precise timings for sections as a second check against grossly deforming his works. He believes his music to be more delicate with regards to timing, so one wrong move and snap. Mahler, on the other hand, gives loads of instruction with regard to tempo, some metronome markings, but a degree of sturdy flexibility to his music. It seems that his approach to harmony and orchestration further supports this.
As for Mozart and Beethoven, and my answer ("not much"): the two occupy a close relationship in time and practice. I don't want to suggest in any way that this means that their music sounds similar. This is not the case to all but the most basic unacquainted listeners. But they occupy a similar placement in history, and it is logical to see that their music, as performed by their contemporaries, would have received somewhat similar approaches to performance. By that I mean that words like Allegro and cantabile still meant roughly the same thing to Mozart's contemporaries in the 1780's as it did to Beethoven's in the 1820's.
The yawning chasm between the two is the abundance of additional performance instructions that Beethoven gives. Remember that not until Gabrieli did we even have the innovation of notated dynamics but even Mozart did not fuss too greatly with them, frequently limiting himself to simply p and f with fewer marked crescendi and diminuendi, while Beethoven added more of these, including the more frequent use of pp or ff. (Caveat: Mozart uses almost all of these markings some times, but it is a question of frequency.) This process of specificity exploded further by the time of Tchaikovsky and still further with Mahler. But does that mean that we do not want to make a distinction between different forte sections in Mozart, or shape the phrase? I say no. It is simply that Mozart generally left it to the performer, familiar with the prevalent style, to know what would be appropriate, while Beethoven explicitly asks the performer, lest there be any doubt. All of these comments map almost perfectly onto their use of expressive tempo markings. Mozart tends toward the implicit, Beethoven toward the explicit. In fact, it is even more true in the case of tempo.
Take for instance, your examples. It's hard to make a direct comparison because of the number of variables. What would be easier would be to compare opening movements in the genre that begin in the same key, same time signature, and same tempo marking. But just to quickly compare, KV331 opens immediately with a stable tempo that is maintained until variation V, but with no indication as to how or when to change tempo. Beethoven, on the other hand, starts Op. 31/3 Allegro, but can't even make it three measures without giving directions for a change in tempo! So you see that there is a wide diversity between the musical style of the composers in how they manage time, but less diversity in how much we should additionally mold it beyond the directions of the composer. Not that we shouldn't do it, but that we shouldn't do it altogether differently. A comparison of the range of acceptable rubato usage between the performance of Mozart and of Beethoven would show that in both cases, one would be wise to tastefully but very judiciously shape time. The best way to see this is in careful study of many different recordings of the same work.
For the most thorough set of interpretive analyses of recordings that I've ever encountered, look into Gunther Schuller's The Compleat Conductor. It is a fascinating look, microscope in hand, at 8 masterworks. It contains dozens of tabular comparisons between the approaches of different conductors, each table to one very small phrase or interpretive decision, to get a sense of what the range is. Most often, given that tempo is the conductor's most essential domain, a trend emerges as to what consensus, if any, there may be between a range of interpreters. I would love to know if anyone has made a similar effort in covering another genre or other works.
I'll be interested in hearing your response or the response of others. I'd especially be interested in anyone who would like to compare two more similar movements in Mozart and Beethoven along the lines I suggested, to see what more specific truths can be added to this as far as performance approach.