I think that looking at Bradley versus Smart on the defensive end is kinda like looking at Al Horford vs Demarcus Cousins on the offensive end.
Al Horford isn't able to really initiate / create offense. He gets his offence within the system. Most of his points come from open jumpers, or else they come from open looks and mismatches inside. He's a really good shooter and a really good finisher, so when you give him easy looks he almost always makes them - and he's a good offensive player for that reason. However if you don't give him easy looks, it's very hard for him to impact a game offensively. For the most part, he takes what's given to him and he is very effective at that.
Demarcus Cousins on the other hand doesn't hit his open jumpers as consistently as Horford does, and probably doesn't finish around the basket quite as efficiently as Horford does, and he forces things sometimes more then Horford does. As such his shooting percentages are lower, etc. You could argue his game is probably less fundamentally sound then Horford's is. However, Cousins is a guy who can create offence at will, and it doesn't matter who he is up against or how they defend him, he will almost always find a way to put points on the board.
I see the Bradley vs Smart argument the same way.
Smart is probably more fundamentally sound on defence. He gets out to shooters very well, if teams show him the ball he'll strip it and get steals, if he has a size/strength advantage he will use it well, and he's versatile enough to switch on multiple positions in the pick and roll. He's the type of guy who will take advantage of the opponents errors when they happen, and he's really good at that.
Bradley might be less fundamentally sound on defence in some of those regards, but he ultimately creates defence. He doesn't just take advantage of opponent weaknesses when the opportunity arises, he creates opponent weaknesses when there seemingly is none. With his quickness, his aggressiveness, his length and his incredible motor, he will get up opposing guards all day every day.
With his constant pressure, he:
1) Makes guys, who are otherwise supremely confident, start to second guess themselves
2) He makes guys, who are otherwise supreme athletes, tire and get worn out
3) He forces guys, who are excellent shooters, into extremely difficult shots
Smarts an outstanding defensive player, but I consider him to be more of a passive defensive player. He mostly plays quietly effective defence in the background, and strikes when the opportunities opens up. Bradley is more of an active defensive player because he is impacting the game defensively, on the ball, pretty much ever second he is on the court.
Advanced stats undoubtedly show that Smart has more of a positive impact on defence. But if you ask opposing team's star players, they seem to fear Bradley more then maybe any other player in the NBA defensively. Guys don't want to play Boston because they hate being defended by Bradley. When you can get into a star's head like that, you have essentially won the battle before the game has even started.