The Vision's powers in the comics are, according to Comic Vine (I've added emphasis to his computer-like capabilities):
The Solar Jewel on the Vision's forehead absorbs ambient solar energy to provide the needed power for him to function, and he is also capable of discharging this energy as optic beams; with this, he can fire beams of infra-red and microwave radiation. In extreme cases he can discharge this same power through the Solar Jewel itself which amplifies its destructive effects considerably, albeit at the cost of losing most of his resources. The Vision also possesses the ability to manipulate his density, which at its lowest allows flight and a ghostly, phasing intangibility, and at its heaviest provides superhuman strength, immovability, and a diamond-hard near invulnerability. The Vision is capable of reaching a density ten times greater than that of depleted uranium. On one occasion, the Vision uses this extreme durability to render unconscious the villain Count Nefaria. The Vision often uses his ability to alter his density against foes, by phasing an intangible hand through them and then partially re-materializing it—a process he describes as "physical disruption." This effect typically causes great pain and results in incapacitation, as is the case when used against the Earth-712 version of the hero Hyperion. Being an artificial life-form/android of sorts, the Vision possesses multiple superhuman senses—as well as superhuman stamina, reflexes, speed, agility, and strength (even without being at high density). The Vision was trained in unarmed combat by Captain America, and is an expert in the combat use of his superhuman powers. The Vision is an expert on his own construction and repair, and is a highly skilled tactician and strategist. The Vision controls his density by interfacing with an unknown dimension to/from which he can shunt or accrue mass, thus becoming either intangible or extraordinarily massive. The Vision also possesses superhuman analytical capabilities, and has the ability to process information and make calculations with superhuman speed and accuracy.
That's a lot of powers, but very little of it comes from his
positronic synthezoid brain. So it seems that The Vision can think like a computer, but can't really act like one in the same way that Ultron can.
On the big screen, Vision's computer-related abilities seem to have been dialed up a bit. As you mentioned, in Avengers: Age of Ultron he was able to hack into Ultron's operating system. We don't know much beyond that, however.