The evolution of the concerto form (for all instruments) follows the evolution of the musical forms in general. For example, the Classical period favored structure, intellectualism, rationalism, and economy of means; the Romantic period, which closely followed the Classical period, is the direct antithesis and reaction to the relative constraints of the Classical period. The hallmarks of the Romantic movement are emotion over rationalism, freedom over structure, more versus less, etc. In your examples of Mozart and Tchaikovsky, when listening, can you hear the smaller orchestra, the structure of the musical episodes (sonata allegro form), the limited range of the piano, and the relative compactness of his keyboard writing? Tchaikovsky, on the other hand, is grand, dramatic, much more contrast in dynamics, longer, and the structure (although still following the traditional sonata allegro form) is less obvious and less necessary. Remember also that, through natural human and industrial evolution, our lifestyle and social norms change, too. Think of how different we are today vs. 1912-1962.
In terms of performance, depending on how you define virtuosity, the piano writing and performance was no less technically demanding in the Classical (or even Baroque) than in the Romantic period. (Think again of an exposed ankle during Downton Abbey's days, vs. a string bikini today.) However, the soloist was still very much a part of the ensemble (for the most part), in that the interplay between the soloist and the orchestra was very much supportive of the other, rather than against each other. In the late Classical and most of the Romantic period, the idea of the struggle and drama of David vs. Goliath was more apparent. It's almost as if one were fighting the other, with the goal of vanquishing the other. And, as the previous poster pointed out, the soloist became the star of the concerto, and Liszt's long hair and dramatic mannerisms became part of the public's image of a concerto performance.