There is the simple issue that Mike was doing too many things at once for it to be Prof - he was the phone system, allowed null numbers to be routed to him, and was at the very least Adam Selene - holding joint conversations with Prof in the room, which would have required a perfect anticipation of everything the others might do or say for entire nights at a time.
There are also the incredibly complicated calculations, the fact that he was holding hundreds of conversations at once as Adam Selene, and the programming. When The Moon is a Harsh Mistress was written, computer programming took an incredibly long time in any format, and Manny explains some of these difficulties in terms of programming - for example - the rock strikes on earth. He gives Manny print outs of thousands or programs that the 'lesser' computer operating the secondary catapult might need - and which Manny would then have to key in by hand.
It's explained that Manny is the best computer man in Luna - and therefore cheaper than bringing up an expert from Earth every time was needed. Additionally, as a precursor to all of that, Prof was teaching him advanced math and eventually they ended up even with one another before Manny pulled ahead. In essence, Prof just wasn't good enough at math or computers to do a lot of what Mike does in the course of the novel - and the novel regularly points out the varied and diverse uses of the "self-aware" phone system and the ability to talk to many people at once, something Prof definitely couldn't do. And what Mike does do would be the life's work of a team of 1,000 mathematicians, scientists, political planners, marketing specialists and techs. Prof's good, but no one is that good.
Perhaps in modern times, it might be possible to program a system that would act like an AI without actually being one - but I doubt Heinlein ever even really thought of that. He lived - and died - long before computers could do more with graphics than make banners and homemade birthday cards with very pixilated, pre-selected graphics, not the modern CGI which has gotten so real it's created the idea of the uncanny valley. When Moon was written, computer programming was in its infancy - and the concept of a 'yes, no, maybe' computer was a brilliant but completely unrealistic idea of how computers would be made more flexible and able to "draw correct conclusions on limited data" (what Mike is designed to do). In essence, for Mike to have really been Prof would have required a greater suspension of disbelief when the novel was published in 1966 (or even when the author died in 1988) than that the computer was a full AI. And in some ways, the novel can be read in such a way that the rebellion is only the backdrop for Mike's story.
