Some bikes' two wheels are farther apart from each other than other bikes. What difference does it make practically (riding-wise)? When is one preferable to the other? Are the ones with longer distance in between more stable and good for relaxed riding?

Shorter wheel base for racing, longer wheel base for touring. The longer wheel base bike will be more comfortable in the long run. A longer wheelbase bike often has a more relaxed head tube angle, increasing steering trail and stability. Trail has a greater effect on steering than wheelbase.

Thanks--makes sense.

In addition to the above comments, bikes with a very short wheelbase usually have no room for larger width tires or fenders and panniers on a rear rack may interfere with the rider's feet.

That last depends on the style of bike. Mountain bikes often have very short wheelbase lengths that are much the same or even shorter than road racing bikes. Yet they have room for humongous tires in some cases and "merely" big tires in the rest. Mind you mountain bikes get around the fouling of the feet by using a smaller wheel size.

Overall though I've found that all else being equal a shorter wheelbase handles slightly snappier than a longer wheelbase. Even a little difference is noticable such as when trying out two different sizes of the same brand and model.

If you're gonna carry big panniers on a MTB, as a general rule, you'll want to look at pre-1986 MTBs. That's about the time many of them stopped sporting limousine rear triangles.