English Translation
Bob enjoys playing sports.
What does “treiben” bring to the party that “spielen” does not?
“treiben” is the word (besides simply “machen”) that’s usually used together with “Sport”. It’s a collocation. “spielen” is only used together with kinds/disciplines of sports, e.g. tennis, basketball, etc.
I think the two sentences don’t exactly match in meaning.
In german, “sport treiben” emphasises the workout aspect, i.e. staying fit by physical activity, and is less about which kind of sport it is. It may just be going to the fitness studio. Whereas the english “playing sports” to me sounds like “participating in a sports discipline”, i.e. one that can be “played”, and which would exclude a gym.
TLDR: “spielen” and “Sport” don’t fit together.
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English Translation
Bob enjoys playing sports.
I had previously encountered the German ‘sport’ more as exercise than a particular (English) sport
That would be “Sportart”. 
Soccer is a “Sportart”, same a Basketball.
But playing soccer/sports is translated with “Sport treiben” or “Sport machen”. Germans don’t say “Sport spielen”. You use “spielen” with a “Sportart” (meaning: particular type of sports), for example “Basketball spielen”, but not with “Sport” in general when the “Sportart” isn’t specified. Then it’s “machen” or “treiben”.