This is a difficult question to answer, because single coils themselves don't all sound alike by any stretch. It depends on the magnets, wire type, overwinding, etc. So, any given humcanceling J-bass pickup might sound like some single coils, but not others.
That said, if you're looking for a noiseless version of something like stock U.S. Fender singles, I think DiMarzio's Area J are great. It's classic, straightforward J-bass tone IMO. I also use Lindy Fralin's Split-Jazz, which I prefer due to their thicker lows and slight extra growl. They seem to have more "character," though of course that's tough to define, and how they sound depends on the bass too. Really, there isn't a huge difference between the two, and IMO both sound very close to some single coils.
So, what do you lose? Normally, a little "air" in the highs, but if you roll off your treble a bit anyway, as I do, you might not notice. Some noiseless designs seem to hype the treble to compensate for this, which IMO sounds a little artificial, but some might like the modern sheen. You might also lose the "open" quality in the lows you'll find in singles, but this is more subtle IMO.
I also think singles have a more instant response, which I think some call "bark," that for whatever reason gets slightly reduced in hum-canceling designs. This is almost more of a "feel" issue than a "tone" issue, and at low volumes I doubt I'd notice. But after years of playing noiseless J-bass pickups only, I noticed it when I hit the stage with my MIM J loaded with U.S. Fender singles. Again, this is subtle, because the Area J and Fralins do have some bark -- the pure singles just seem to have a touch more.
(Insert all IMO and YMMV type qualifications here...

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