This always puzzled me. I mean Pokemon X and Y let you customize your trainer's wardrobe, but only to swap things in an out. You're not capable of going without shoes or a hat.
A friend of mine told me that it aids in swimming, like you swim faster, but I've never thought to try it. The swimming feature is honestly the only reason I can figure that this is even a thing New Leaf allows, whereas City Folk and games past didn't.
I don't think there is anything different when you take off your shoes. It's more of s personal preference though. I don't really care for any of the shoes in the game. I end up just wearing sandals for my character. Although the sound of shoes versus bare foot is fun to test out. I like the sound of bare feet rather than shoes sounds.
I just wanted to mention how it's pretty neat that the developers put in separate sounds for when you're wearing/not wearing certain types of footwear. I noticed during a small rain storm yesterday that when I walk in the grass, it sounds different from when I walk in the mud. I also noticed that when digging up a hole, even in a grassy area, mud came out instead of just dirt.
The developers just did a really nice job of implementing these tiny details. I like it a lot~
Different kinds of shoes actually make different sounds? Wow. I really haven't noticed that before. And I've barely ever gone barefoot so I haven't noticed the sound (or lack thereof) either.