The breakout box moves the A/D converters outside the computer and is also a bit more convenient. The input/output connections are on a separate box which can be rack mounted, leaving just a single wire going between the breakout box and the PCI card in the computer.
As Dacha said. The break out box brings all of your TRS (in the case of 1010, Delta 44, and Delta 66) jacks upto your keyboard, desktop or wherever it's convenient. Be aware the Serial cable (the cable that connects your break-out box and your PCI card) is not that long, so you cannot put it in the next room.
It's a hell of a lot better than crawling under your desk to plug your stuff. Granted you could buy the 1010LT and buy a patch panel (which would work as a breakout box essentially) and put it on your desk. The only thing by using a patch panel, it will add some line noise (depends on the quality of the patch panel) and it's an additional cost.
I was looking at the endings on the 1010LT, and how do plug that into my mixer? the endings on the cables are different, whats the name for these endings?