Though they were all highly-developed and prolific composers from the Baroque eraThough they were all highly-developed and prolific composers from the Baroque era

Though they were all highly-developed and prolific composers from the Baroque era, the three seemed to explore their own respective specialties in ways that made them very distinct.
Bach is known for his highly developed contrapuntal style. In this way, he is truly peerless and innovative. He is known as a master of the fugue that none have ever surpassed. His well-tempered klavier exhibits his mastery of the keyboard and contrapuntal writing on the keyboard. He was also deeply experimental, and his writing is highly virtuosic; Examples include his third solo violin sonata, BWV 1005, where he manages to write the largest fugue of his lifetime for the solo violin, an instrument scarcely considered a polyphonic instrument at the time. He also wrote the Chromatic Fantasia and Fugue, BWV 903, which took the fugue to another level of musical of expression.
However, he was most known in his lifetime for his sacred music, and his strong Lutheran faith. He wrote hundreds of Cantatas and oratorios, but never wrote a true opera like Handel did. His themes are riddled with secret numerological devices and Lutheran hymns.
Handel, on the other hand, wrote music that was clearly less religious, writing many important operas. He worked frequently in Britain, and his usage of counterpoint is impressive, but not at Bach’s level. He uses many of the same structures as Bach, but has a penchant for stacked harmonies that avoid counterpoint more. In this way, his music is designed to clearly articulate a dominant voice, accompanied by non-dominant voices (homophony) whereas with Bach all voices in a piece are treated equally, like a conversation where everyone speaks over each other.
Vivaldi’s style, however, is highly distinct from both, and exemplifies Italian homophony to a much greater extent. You never lose the melody with Vivaldi. His specialty was the concerto, of which I believe he wrote more than any other composer in history. Apparently, each of his concerti were written for different children in the orphanage where he was a minister. His music is often written in stacked harmonies with a unified rhythm, paving the way for classical harmony. He almost never wrote vocal music, compared to Bach and Handel.
These are all blanket statements, and you can find instances where Bach takes ideas from Vivaldi, and Handel from Bach, and so on. However, just listen to them all and you will hear it for yourself over time.