Are fresh baby carrots better than full sized carrots?

Same. The "baby carrots" sold in stores are regular, fully-grown carrots cut into those little shapes. They're not "baby"-- that's just a marketing term. If buying carrots cut and clean and in little shapes makes it more likely that you are your kids will eat them, go for it!
Not necessarily. "baby carrots" are actually regular carrots which have been trimmed to resemble baby carrots. There is no health difference but there may be a convenience difference.