Biggest difference between those two bikes is that the Trek has a suspension fork while the Motobecane uses a rigid fork. Given that the Motobecane has an aluminum frame, it's going to be a bit bumpier than the Trek. But those 40mm tires will cushion bumps. Shouldn't be a problem on streets or smooth trails.

I'm not quite sure what a Shimano "button" shifter is, so I have that concern about the Motobecane.

If you go with the Motobecane, make sure you know what size is the best fit for you. And unless you are very handy with bike adjustments, you'll probably have to pay a bike shop to finish the set up for you.