A few thoughts on the prospect of Mourinho as United manager.
Transfers
One of the major advantages of Mourinho is basically his fame. Having managed Real Madrid and Internazionale (and, to a lesser extent, Chelsea) in recent years means he is would be quite a glamorous appointment. The current generation of top players have played in an era in which Mourinho has won trophies in a high profile way. We can reasonably speculate that they are thus somewhat more likely to commit the next few years of their careers to United under Mourinho's leadership than they would be under other, less established candidates. We need to strenghten in at least three positions this summer and Mourinho's fame might help us attract higher quality players.
The Rooney question
One of the most important tasks for reshaping our team over the next year or so is phasing out or selling Wayne Rooney. For Moyes the prospect of managing a player with the stature of Rooney was a dream come true, and instead of taking the opportunity of moving him on that Ferguson had (I believe) purposely engineered, he chose to make him more important than ever to our team. Van Gaal has seemingly embraced Rooney too; it's hard to do anything but speculate about this, but my guess has always been that rather than blind faith that Rooney was a superb player, Van Gaal's positioning of him at the centre of his plans about accomodation. He arrived a club which was seriously lacking in defence and midfield, and found Rooney near the start of a long and record breaking contract, and basically decided there were bigger fish to fry. The situation has now evolved to the point that Rooney seems to present a blockage in our first eleven. Van Gaal could have used his arrogance and authority to address the Rooney question; Mourinho would also have this chance. Of course, he could duck it and decide his priorities lie elsewhere, but compared to a more timid, less experienced manager there is a greater chance he'll act strongly on this.
Psychology
Normally I roll my eyes whenever I hear people claiming that the problem with their team is lack of passion (it's the standard refrain of the clueless casual England supporter), but our team has looked depressingly timid in recent times. I think that the drive that you saw in classic United teams is to a significant extent a function of self-belief. We don't seem to believe that we can rise to the top and we look defeated when things go wrong. What makes teams overcome adversity is belief in themselves personally, belief in their teammates and belief in the plan. At points in his career Mourinho has shown that he is capable of inspiring this kind of belief (notably in his first spell at Chelsea), so there's a decent chance he'd be able to stop the rot here. On the other hand he has also shown the capacity to entirely undermine team spirit (see his second spell at Chelsea and the Carniero debacle), so it's by no means guaranteed.
Style of play
Moyes' and Van Gaal's conservative tactics have been a source of intense frustration. One of the things that made Ferguson's teams far more enjoyable to watch was that he was an extremely skilled risk-taker. Mourinho is not a risk-taker in the same way and has tendencies towards conservative tactics. He famously believes that big games are won by the team that makes the fewest mistakes, so in games against, say, City, we would probably have to get used to the spectacle of United playing with nearly everyone behind the ball, waiting for City to make a mistake.
Barcelona psychodrama
The presence of Guardiola at City makes Mourinho into even more of a wild card than he might otherwise be. The fact that Van Gaal has a playing philosophy seems to have been a source of consternation to many posters on here, but the same is true of Mourinho, who seems to have deep psychological issues with Barcelona, which manifest in his desire to demonstrate his superiority to tiki-taka, Guardiola etc. This has led to some questionable tactical choices, and the fear is that this fixation means United becomes the latest stage for him to seek vindication of his own superior tactical acumen, clouding his judgement.