Why are Fallout's computers so backwards?

As far as i know, there is no logical explanation behind it. The game itself is rather old, with the first part being released 1997. I'd say they never had a reason to actually change the theme.

Wikipedia says:

Although the series is set during the 22nd and 23rd centuries, its atompunk retrofuturistic setting and artwork are influenced by the post-war culture of 1950s America, and its combination of hope for the promises of technology, and lurking fear of nuclear annihilation.

Out of universe, Fallout is a "retrofuturistic" setting: the setting is based on the future as imagined at some point in the past (in this case the 1950s). The computers are just a small part of that.

Other common uses of this device include steampunk, so this 1950s future is sometimes referred to as "atompunk".

Possible speculation: higher-yield powerplants in vehicles and in the power grid from extreme nuclear utilization reduced the impetus for miniaturization that led to the computer age. If you look at very old electronics, vacuum tubes and all, and look at new electronics, you notice something very interesting. In general, older electronics use a lot more input power for the same output, and they are usually considerably heavier.

Weight is an issue mainly when you transport things - in gasoline powered cars, extra weight reduces fuel economy. If you're driving a nuclear-powered car however, the economy loss for heavy equipment is probably minimal.

As said, older systems tended to use more power. Again, not a problem if you're putting nuclear reactors everywhere. You can see this especially in older televisions and radios. No incentive to economize.

Lastly, one possibility is electromagnetic interference. See, when a small reactor goes boom, just like when any nuclear bomb goes off, there's significant EMP output, that can damage electrical equipment for miles around. Some older, tube-type (or "analog") equipment was more resistant to EMP spikes than modern, solid-state equipment. Since nuclear power was apparently in every household, maybe that was a concern? I know high-altitude EMP tests (Operation Fishbowl, of which Starfish Prime was the largest) concluded that a high altitude explosion of relatively small bombs could EMP thousands of miles of surface area.

Just some thoughts. The possibility we're overthinking it a bit exists. But I think some of these idea hold a teensy bit of water