The bottleneck is indeed very slight. SAAMI spec for a 44-40 chamber is 0.4710 diameter at the base of the cartridge, and 0.4435 at the mouth. That's a difference of just .0275, or 1/36 of an inch. By comparison, the 44-40 and 44 Remington Magnum case are almost the exact same length (1.248 from rim to mouth). The 44 mag, however, is 0.4600 diameter from the rim to the neck, i.e., it is "straight" while the 44-40 narrows from .4710 down to .4435.
The 44-40 can be a challenging cartridge. The thin case mouth means it is a little less forgiving of some loading errors, such as undersize expanders, oversize bullets, not enough bell to the mouth, etc. The biggest problem, however, comes from the variability in the firearms it is chambered for. In the old days, 44-40 firearms had .426 +/- .001 bores, which called for lead bullets sized to .427. At some point, however, many manufacturers started putting 44 spl/mag barrels on their 44-40 firearms. Colt, for instance, offers dual cylinder Single Action Army revolvers chambered in both 44-40 and 44 Special. Since the 44 Special/Mag barrel is a slightly larger diameter, that's the one that got used. Even in single chambering revolvers, however, many manufacturers now put 44 spl/mag barrels on ALL their 44-40 products.
Since the 44 spl/mag barrel is .4295 in the grooves, you need .430 or .431 bullets for best accuracy. Fitting .430 bullets in 44-40 brass, however, can be quite a challenge. 44-40 expanders are usually sized for the smaller .427 bullets. Modern brass is also thicker than the old days. Even if you use the correct expander for .430 bullets, you may find that the resulting 44-40 cartridge is more hourglass than bottleneck, and may not fit in a SAAMI-spec 44-40 chamber.
The only real reasons to prefer 44-40 over 44 special or magnum these days are: (1) authenticity, if you're a cowboy shooter; and (2) black powder. The 44-40 round was introduced in by Winchester 1873 (as the .44 Winchester Central Fire, or 44WCF) and, along with the 45 Colt, is the round that won the west. It is also an excellent black powder cartridge. The thin neck ensures it will expand and seal the chamber better than any straight walled case, which keeps all that fouling from blowing back into the action and jamming up your rifle. The slight bottleneck also eases extraction when chambers are dirty. Both of those factors helped make the repeating rifle successful. Winchester knew what it was doing.
I load thousands of black powder 44-40 cartridges every year, as I compete in the cowboy action shooting black powder categories. For general revolver shooting, though, the 44 Special and magnum are better choices.