First off, just so you're clear, there are actually two versions of the SB-1. The first was only made from 1982-84 and features a slab body (no arm or "tummy cut" contours) and a single coil J-style (though slightly larger dimensions) in the P-bass position "sweet spot." The second style (often referred to as "second generation"), introduced in '85, is a more traditional P-bass style with traditional body contours and a split-coil pickup. For the purposes of this comparison, I'll assume you're interested in the 2nd generation model.
Two major differences, really:
1) With very few exceptions, the SB-1 features a G&L patented Magnetic Field Design (MFD) split-coil pickup which is hotter and more aggressive than a typical P-bass pickup (though this can be tamed with simple manipulation of the volume and tone controls). The MFD has adjustable pole pieces. The exception to the rule is that the first of these basses produced (from 1985 to early '86) feature a Schaller PBX pickup (virtually identical to today's DiMarzio Model-P) instead of the MFD. The Schaller also features adjustable pole pieces but otherwise looks like the traditional, boxy P pup whereas the MFD has a more rounded profile along the top if you look at it carefully. The Schaller is a hot pickup as well, but closer in sound to a vintage P than the MFD. The LB-100 features a vintage-style Alnico magnet split-coil manufactured by Gotoh.
2) The body profiles are different. The SB-1 is somewhat downsized from the original P shape and slightly thinner front-to-back, whereas the LB-100 is a straight-up P-bass body. The pickguards are also slightly different as a result. The controls are laid out the same (typical volume/tone), but the placement is slightly different.
It's also worth noting that the LB-100 was originally called the Legacy Bass, but the name was changed early on because of a trademark issue with that name. The only difference between the Legacy Bass and the LB-100 (aside from production numbers) is the headstock decal.
Both are great basses. Hope this helps.