My understanding is that there's a huge "con" in that the proper bore dimension is ALMOST the same as the 44Russian/Special/Magnum family, but not quite. 44Mag and it's ancestors need a .429 barrel. 44-40 needs...lesse, I *think* it's .427 but not certain. Do some googling; if you can't find anything, lemme know 'cuz I recall reading about this not long ago...
Anyways. The inevitable problem is that manufacturers set up "44-40" guns that use the same .429 barrels as their much more popular other 44s. Ruger did this with the 44-40 Vaquero. Which means accuracy sucks and leading is nuts unless you understand what you've really got and handload what amounts to a wildcat by mixing the various specs. In other words, .429 bullets in 44-40 cases and similar adjustments.
If you DO get a "true 44-40 gun", then finding loaded ammo is a pain and scoring pre-cast (or jacketed) bullets in the proper bore (instead of the common .429) is difficult.
All these issues CAN be dealt with but do some research first and realize that it is DEFINATELY a caliber you're going to need to handload for unless you're made o' money.
The advantage: it's a low-pressure bottleneck round that drops easily and cleanly out of single action rounds, and has good accuracy potential when loaded correctly.
My opinion: skip the 44-40 and go with the 38-40. The latter shares most of the same advantages but the bore diameter is the SAME as the modern 40S&W and 10mm rounds (.399 - .400" - this "38" is really a 40). Go find a Ruger Vaquero in 38-40/40S&W convertable. These two calibers are actually capable of sending the same bullets downrange at the same speeds...but the 38-40 operates at lower pressure (bigger case). You then use the 40S&W cylinder for cheap practice, or have it reamed deeper to 10mm and use that as your high-power "woods round" and use the 38-40 for SAS/CASS games where it's reload/unload speed shines when you're reloading against the clock.
38-40 is more popular today than the 44-40 and loaded ammo is more widely available.