Both are solid choices and it really comes down to what you need. A 10×12 is a 120-square-foot footprint, and for bikes, yard tools, and a good amount of storage, that's plenty for a lot of people.
Where the 12×16 pulls ahead is flexibility. On a 192-square-foot footprint, you've got room to set up zones: storage on one side, workspace or seasonal gear on the other. Everything fits without feeling like a puzzle. And here's the practical angle: in most cities and townships, 200 square feet is right around the maximum you can put up without additional permitting headaches. A 12×16 gets you right up to that line, giving you the most bang for your buck on a decision like this.
We're not trying to push anyone into a bigger building than they need. We just know from experience that the jump from a 120 to a 192-square-foot footprint makes a bigger difference than most people expect, and the price difference is usually less than you'd think.
