I find the difficult part of professional improvisation is that it requires you to be a complete musician. That means that you know the ins and outs of music for your instrument, you know all the tools and tricks of the trade. Of course anyone can improvise and muck around on the piano, but the improvisation, of say a concert performance, requires a great deal of musical understanding. So much so that pretty much no classical pianists improvise in concert, usually you only see that from the "Jazz/Modern" side in performance(which by no means is any easier than classical improvisation but certainly more popular these days).
I hear the stories of Liszt making women in his audience cry when he improvised in concert, improv back then surely then must have been was quite similar to actual compositions in their structure and clarity. It is hard to say what is a good improv or bad, it is a very subjective thing more so than the subjectivity of what makes good performance of a composition. However I personally feel that concert standard classical improvistaion is a lost art somewhat and only the complete musician can achieve it.
The great majority of professional classical pianists cannot improvise because of the way in which they are wired to play their musica. Most classical pianists are predominatnly slaves to the words (notes in sheets) and can only improve articulating them in a professional manner. They cannot come up with musical talk themselves spontaneously in the style that they are used to speak in. If they do try it never reflects their intention or with freedom of thought. Improv is the final frontier of musical development in my opinion only because I see many great professional pianists who never learn to improvise.
Interestingly enough many composers who are also pianists like to improvize and can do so with astonishing freedom. For example David Foster, a world renouned music figure, can on the spot knit together chords and melody completely effortlessly. The ease in which he does it and the mastery of his understanding of melody allows him to produce countless songs which have been world wide hits. This, in my opinon, is a real gift, given from God or whatever you might think, something outside of ourselves, something that just can't be learnt. These people just live music they understand it so well that it just pours out of them, they don't have to think about it, the decision making is all in the subconsious and the ear controls everything they do.
This isn't to say that those that never improvize or have the inability to are lesser musicians. To be a complete musician you have to be a performer and composer. Some people just like to specialize and thats totally fine.