In theory, the 31.8 bars should be slightly stiffer than the 25.4 and 26.0 bars. It probably doesn't make any difference for most riding, and in some cases, flex is deliberately put into the bars for comfort (Specialized Zertz).

I think there are a few issues.

More 1" stems (both quill and threadless) come in 25.4 and 26.0 sizes than in 31.8 sizes. This means that the smaller bar sizes might be better for working on vintage bikes. However, there are ways to get around this limitation.

Almost all 1 1/8" threadless stems come in 31.8, so working on newer bikes, one should stick with 31.8. Also, the few rare 26.0 stems for 1 1/8" threadless often have 2-bolt faceplates, when a 4 bolt faceplate is likely a better choice.

I have come to like the flat-top "aero" bars. Unfortunately there are very few 26.0 flat top bars available. And they still typically require an open face stem which is rare for quill stems.

I'm not a big fan of "ergo/anatomic" curve bars, but I do like a tight top curve to the bars which gives better brake lever placement. Thus, either "ergo" or "compact". There are a couple available in 25.4 or 26.0, but most of the more modern shapes are coming out in 31.8. Also, if buying used parts, the 31.8 are much more common.

Also, almost all new carbon fiber bars are being made in 31.8, with a few vintage TT bars still showing up.

Anyway, the choice really depends on what you're mounting it on, but if I was building a "modern" bike, I'd stick with 1 1/8" stem and 31.8 bars.