They are both closely contested. You wouldn't go wrong taking either one. But, you asked for one or the other.
Best in Sport: Brazillian JiuJitsu.
BJJ tends to take Judo in sport matches. You can look up the results on YouTube. Judo does win from time to time though. But, generally, it is played by BJJ rules and Judo is at a disadvantage. BJJ just has a better groundgame than Judo. BJJ players spend a lot of time on the ground so, they can last a long time on the ground if need be. Judo only has a relatively short time on the ground in Judo contests. BJJ also has more submissions than Judo. BJJ has leg locks for one thing, they also are far better in the guard position also.
Best in the Street: Judo.
I would say Judo because it is the master of throws. You can grab someone and slam them with a Judo throw and that may be the end of the fight. BJJ has throws and takedowns but, they aren't as practiced and they don't practice as many variations as Judo. Judo also ingrains the idea in you of having a quick ground game. So, if you get a guy on the ground instinctively you work fast to get the submission. In Judo contests they have that short time on the ground so they work fast to get a submission. BJJ takes their sweet time on the ground for a submission mostly. So, that's why Judo is a tad better on the street than BJJ.
Best to Mix with other MA: BJJ
BJJ fits right in with several martial arts because of their heavy focus on the ground game. There are several martial arts that are very weak or have nothing in the ground department like: Karate, Tae Kwon Do, Boxing, Muay Thai, Aikido, etcetra. So, BJJ's heavy emphasis of ground work will work just great with those martial arts and any others that don't have a good ground game. Judo's fine too but, it's main thing is it's throws. Some arts that are mainly stand-up striking still have throws and sweeps like: Karate, Aikido, Kung-Fu, and Tae Kwon Do. Judo's inferior ground game to BJJ's just makes BJJ's ground game more important for those other martial arts to mix with.
All around best MA: BJJ.
I say BJJ because they may teach strikes to go along with the grappling. Judo rarely teaches it's striking game. BJJ's striking is practical oriented and meant for the street. It also has the extra submissions and the best ground game of the two.
So, BJJ wins in most of the categories in my opinion. But, you really wouldn't go wrong taking Judo either. I would say if you don't mind BJJ's heavy ground grappling, then, yes, take BJJ. But, if you don't like to roll on the ground for minutes on end (which I don't. I hate ground grappling. It's very claustraphobic at times) then Judo would be a better choice. But, BJJ has a lot of great things if you can get in it and enjoy it.