The difference is that it sounds like a P bass. If you listen to any song, by any group, since electric bass was invented, about 70% of the time, you'd be hearing a split pickup. Unlike a traditional humbucker in the neck (like on a gibson) or an enlarged one near the bridge (like on a musicman) a split coil pickup in the direct middle of the playing area gives you increased low end, slightly decreased high end, and a very extensive mid range. The positioning also captures a very certain harmonic range.
Traditional humbuckers give off a very honky midrange and kill the highs. They also decrease output.
Single coils lack mid range and low end, and therefore sound very thin. They have increased output, but also have a very shrill high end.
After building the Precision bass, Leo Fender decided that splitting the coil that he had in the bass gave it more of a "bass" tone. It then became the norm, the foundation, and the world renowned standard for electric bass. Yes, there are many different styles and setups now, but the fact still stands true that "a split coil in the sweet spot is an electric fretted bass guitar." The Precision bass. It is still the industry standard.