No personal experience with the .38-40 but in case you don't know, the name of the cartridge is somewhat strange. It is derived from the .44-40 (the original chambering of the 1873 Winchester. The .44-40 was a .44 caliber bullet over forty grains of black powder. You'd think the .38-40 was a .38 caliber bullet over 40 grains of powder, right? Wrong! It's a .40 caliber bullet over .38 grains of black powder. According to Larry Potterfield, anyway:
https://www.youtube.com/watch?v=f8Zd0lZk-B8
My "main match" rifle is a Uberti 1873 chambered in .357. After my wife started shooting with me, I lucked into a Uberti 1866 chambered in .38 Special in a local pawn shop and bought it for her. I had to because she was about to take my '73!
Just for information purposes, the "real" 1866 Winchester was chambered in .44 Henry (same as the 1860 Henry rifle), a rimfire cartridge that is not being made today. According to the Uberti website, their reproduction 1866 rifles are available in .38 Special, .44-40 or .45 Colt. I have heard of other chamberings such as .44 Special and others may be out there.
The Uberti '66 and '73s are similar in several respects and both are fine. Mechanically they are pretty much the same which means they are distressingly stiff out of the box given the price but both slick up great if you will fork up a little more money. Both of our rifles eat ammo like candy. Personally I prefer the '73 but I got such a great deal on the '66 that I snapped it up.
As Hawg Haggen (howdy, Hawg) pointed out, the brass receiver makes the '66 a good bit heavier and it has a different balance point than the '73. Since I bought the '66 for my wife, who is kind of petite, this was a consideration, especially since the one I found is a "short rifle" with a 20 inch octagon barrel with a little .38 caliber bore. The rifle was a good bit heavier than my '73 with the same tube. I ended up having the '66 barrel bobbed to about 17 inches.
One mechanical difference is that the '73 has a lever safety that prevent the rifle from firing unless the lever is fully closed (there is a small spring loaded plunger behind the trigger that is depressed by the lever when closed). The '66 lacks this feature so there is some risk of an "out of battery discharge" if the trigger is pressed before the lever is fully closed. Some of the older (much older) Uberti 1866 reproductions had a lever safety but this feature actually first appeared on the 1873 Winchester rifle. I would not dismiss the '66 on this basis (else I would not have purchased one for my wife) but it's a comparison point.
There are more variations of the '73 available. For example you can choose between a "straight stock" or a "pistol grip stock" on the '73. All of the Uberti '66s I have seen are straight stocked. You can get the '73 chambered in .357 (the '66s are .38 Special) and Uberti offers the '73 carbine in .44 Magnum. Some of the '73 have blued receivers but most have the very attractive "color casehardened" (OK, it's a faux finish) finish on the receiver. The '66s have a brass receiver and some folks really like that so that's a subjective comparison point.
My Uberti '73 is the single most expensive firearm I have every purchased. It has been worth every penny. My wife thinks her '66 is the cat's whiskers. You can't go wrong with either.