In his first meeting with Sherlock, had he acted more like a normal professor (im sure he can act as one) and just tell Sherlock off, may be call him a lunatic trying to seek trouble and deny involvement.
This might put Sherlock off slightly and he will have no chance to get the decoding book which leads to no engagement in the game which leads to auto win for moriaty.
He's a narcessist. He believes himself superior and wants everyone to know it. It would be quite a blow to his ego to weakly cover up his true identity to the detective and instead relishes in it.
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but there is nothing ego damaging to his reputation, he could act like the professor he is and shown signs of intelligence without giving the whole game away.
sharebecause Moriarty knows Holmes won't stop at a simple facade...the very fact the brightest mind of the era came to his university to talk to him pretty much told him he knew all about his activities, so why bother denying it?
Plus, don't underestimate the boiling hatred he has for Holmes, which is fantastically conveyed by Jared Harris's subtle expressions and voice. You can hear his tone almost breaking up a few times. If he could have killed him with his bare hands right there at Oxford, he would have done it
Its cause of the ego, both Holmes and Moriaty are in a league of their own, plus Moran told Holmes the prof wanted to meet him so it was a chance for both to "size each other up" as they consider each other "worthy opponents" but are at odds with conflicting goals. Even if not in direct conflict both would challenge the other based on the fact they have similar minds and huge egos.
It boils down to "there can only be one" as they are both cut from the same cloth.
"Always two there are , a master and an apprentice"