18v allows more headroom but this probably needs a bit of explanation. It's a bit like the difference in bandwidth between a 1mb connection and an 8mb connection and the difference all depends on what you want to do.
If you imagine you've got the vibrations of a plucked string on your PC as a graphic waveform a bit like this:
With a louder the note the peaks are at higher and lower extremes. For the attacky bits in a note, the sound would be relatively loud and show up as sharper peaks. A waveform for a series of plucked notes might look something like this:
What 18v does better than 9v is pass on the extremes of those attack peaks (also known as transients). The extreme peaks normally need a bit of extra oomph to get passed on which the 9v can't do. Transients can also ruin speaker cones and blow HF horn fuses if boosted too much. However in most live gigging situations the difference between 9v and 18v probably isn't going to be that noticeable to fingerstyle players. For people who slap or tap and want the attack of their playing to be passed on faithfully to the amp or PA, 18v might make things sound a little bit cleaner and crisper, especially in a recording situation. It also depends on the instrument, I'd imagine you'd be able to appreciate a 18v preamp a lot more easily in a Status Stealth than in a Gibson EB-3, just because the Status is a much more efficient instrument (although let me add not necessarily better sounding to everyone).
Jonas Hellborg had a 36v preamp in his signature Status bass because he wanted to pass on as much of the transients of his playing as possible. I think he has a similar number of batteries in his Abe Wechter acoustic and it also drove the development behind his Warwick line of amps. I guess this means that in a quiet setting you'll get more of the power, more of the intimacy of someone's playing if they're doing solo work, but I can't imagine anyone ever needing a 36v preamp or being able to appreciate the difference in transient attack when playing in a band.
Some of the EMG circuits (and IIRC the John East preamps) are capable of taking anywhere between 9 - 24v. It all depends on what you're going to use the bass for and what other gear you're planning to use. Its important to think of the bigger picture as well. Its all very well having a bass that is capable of making sweet jesus love to your ear drums but there's no point in plugging it into an amp that is designed for bog standard fender type basses. The benefits are unlikely to be passed on all the way to your speaker. (And speakers are yet another thing entirely that I'm sure AlexClaber will pick up on).