He put unneccessary emphasis on capturing Stalingrad because of a personal duel with Stalin. If he ignored the city and went for the jugular at Moscow he could of well captured Stalin and the whole Bolshevik leadership! iirc Panzers got very near Moscow, 20 miles or so!
Also not releaving the Kessel relying on Goering to supply the trapped troops even though it was impossible for the Luftwaffe to manage leading to the destruction of the (6th army?).
Not listening to generals advice.
Declaring war on the USA.
Mixing some things up, here... Reconaissance elements did get within sight of Moscow, but that was 1941. The drive on Stalingrad was in the 1942 campaign, after the Russian counteroffensive had driven German troops back far from Moscow.
The drive on Moscow in 1941 was not stalled because of Stalingrad, but because forces were diverted to close the cauldron at Kiev, which destroyed most of the remaining Soviet Western army and netted the Wehrmacht hundreds of thousands of prisoners. Perhaps a mistake, but it would have been hard to miss such an opportunity ... and could they have taken and held Moscow if that Army Group at Kiev had remained intact?
As to the Kessel at Stalingrad - he did attempt to relieve it. Manstein and Hoth came within 30 km IIRC, but couldn't manage the rest. The mistake was in not allowing Paulus to break out to the west to meet Hoth.
Now here's the silly question. What would capturing Moscow have done?
It would have brought into Hitler's hands the leaders of the Soviet Union, who apparently never left Moscow even when Panzers were within 20 miles of the place.
Hardly. The archives and a lot of the bureaucracy had already been evacuated and it wouldn't have taken Stalin and the Politburo long to follow if necessary. They were in no imminent danger of capture - only reconaissance units had gotten so close as you state, and there were still static defences (tank traps and the like) to surmount before taking the city.
Moscow was a major railway hub; without it, units to the west (in Novgorod, St. Petersburg, etc.) would have been cut off from the east. It also could have had political ramifications for the Russian leadership, but one needn't engage in speculation to see that its capture would have, at the very least, been a major strategic benefit for the Germans.
As Gustave said.
Also, precisely because Moscow was the railroad hub, the Soviet winter counteroffensive would have been much more difficult to stage if Moscow had been taken.
To the OP: somewhat whimsically, I submit Hitler's greatest single military mistake was allying with the Italians...

. Mussolini pulled Hitler into first North Africa and then the Balkans, diverting resources and delaying Barbarossa, while never being of much help otherwise, except in tying down British naval units in the Med.
All else being equal, who knows how Barbarossa would have turned out if it had started 4 weeks earlier, as planned, and with even more resources which had been squandered in unnecessary campaigns to the south...?