I have Lee dies for .357, and purchased a Lee .38 Spl expander die to allow them to be used for both calibers. I have many other Lee dies for numerous handgun and rifle cartridges. I also have Hornady dies in several other calibers, including .44 Spl/Mag, .204 Ruger, .222 Rem, and a few others. The quality of the Hornady dies is very high, and they seem to be just "that much more" than the Lee's, and I really like their bullet seating guide, especially for rifle reloading. That being said, in practice, the Lee dies do just as good a job, at a lower cost (for some cartridges, such as .358 Winchester, the Lee dies are MUCH lower in price). Also, I highly recommend the Lee Factory Crimp Die, it just turns out such nice, consistent ammo. All in all, I highly recommend that you get a Lee 4-die set for .38 Special, it will allow you to load both .38 Spl and .357 Magnum with only minor differences in setup, and it includes the FCD, which you will, likely, come to believe is a must. It will be similar in price, or less than, a Hornady 3-die set. I do love my Hornady dies, but Lee offers the best bang for the buck in reloading dies. BTW, Lee's internal machining (where it really counts) is excellent, as is the machining on Hornady's dies. The major apparent quality difference is in the exterior finish, where the Hornady dies really stand out. Oh yes, as to use in a progressive reloader, all the Lee dies built in the last 7-8 years are longer than the old ones, and will work well in any brand progressive reloader (I use them in my Hornady).
With regard to your actual question, which is of higher quality, I have to give the edge to Hornady (actually, they are nice to just handle and appreciate the workmanship, almost like fondling a favorite S&W revolver). The cost-benefit race, however, goes to Lee. Either will allow you to turn out high quality ammunition, and will last just about forever.