What's the best route to get to Show Low, Arizona from Phoenix. I can either go through Payson or Globe.
By "best" I mean which route is faster or more scenic?
Google Maps shows your two proposed routes as having a less than 5 mile and less than 10 minute difference. Which one is "faster" becomes almost entirely irrelevant at that point.
So we're left with "more scenic." Of the two routes, one (via Payson) runs almost entirely through Tonto National Forest and Sitgreaves National Forest, while the other (via Globe) is almost entirely outside the national forest boundaries.
Routes passing through national forests are highly likely to be scenic, or have easy access to scenic overlooks, points of interest, etc. The other route which skirts the national forests didn't look all that scenic from a couple of quick peeks at random points on Google Maps.
Though, if you're stopping a lot to take in the scenery, then that will certainly slow down your trip!
So I would say, if you want a fast, scenic drive, go via the Payson route, and if you want a fast, boring drive, go via the Globe route.