While I generally agree with what he says, I would take one issue with the Meadauto answer. Ideally a "remanufactured" item would use all new parts except the case, but in practice this is not always the case. Also, many shops use the term rebuilt and remanufactured intechangeably.
During the remanufacturing process, the unit is totally disassembled, all parts cleaned and inspected, then put in parts bins for reuse later. Parts are checked for obvious flaws, broken gear teeth, etc - but those that are within specified dimensions and do not show severe wear may be cleaned up or reground to come within spec, and shafts with gouging or bearing slippage wear may be built up by welding and remachining. The company may also buy transmissions from junkyards and recycled parts dealers to make up stock for the more commonly damaged or worn parts.
In short, the parts ostensibly (depending on how much you trust third-world or low-budget remanufacturing shops) are serviceable and to specified dimension, but many may NOT be new, and how well the operation adheres to original specifications depends on the diligence and attention to quality of that particular company. One would presume that a transmission that was remanufactured by the original manufacturer would use more new parts and pay closer attention to specified clearances and dimensions.
Also - do NOT assume it has been checked and assembled by an ASE certified mechanic - in fact, I would guess that would be more likely for a rebuilt unit, because remanufacturing is normally done in a factory setting with assembly line workers, not mechanics. I once had occasion to make ann inspection visit to one of the world's largest engine rebuilders (in the midwest) to inspect a large engine that was being rebuilt. In an engine rebuild/remanufacturing plant of over 500 workers, only one was a certified mechanic - the rest were machinists, tool and die men, machine operators, assembly line workers, technicians, etc. This is just like the original assembly line - the assembly workers are normally not mechanics, just machining and assembly workers.
Naturally, the longer the transmission model has been out of production, or the less common it is in the US, the more likely that the remanufacturing process will reuse old parts rather than new as new parts may not be available.
Because a good portion (a third to a half) of the total job cost will be for the labor and sho charges to remove, install, and test/adjust; unless you are very budget constrained or do not intend to keep the car indefinitely I would emphatically recommend the new transmission. It may cost 25-50% more, but will give you a lot more peace of mind and even though the other choices may have warranties, they are likely to be a lot more hassle to get coverage from, may not cover full repair cost if the transmission fails in the warranty period, may be prorated, and may not be accepted at the nearest repair shop you end up if it fails. Having a warranty that requires you to tow the car a couple hundred miles to the nearest shop that accepts the warranty is no fun, and a lot of $ and lost time.