FYI - your "new" set of tongs are the "large" tongs and your "used" set of tongs are the "small" size tongs. The size of tong required (i.e. small or large) is determine by the cartridge/die set. Notice the difference in length of the tong part where the dies screw in to. The chart that is referred to should indicate the tong size required for the cartridge/die set. There should also be an "adapter" that screws in to the tong - usually included in the die sets but often "lost" over time. This adapter is sized to the cartridge that is being loaded. Many are interchangeable - example would be the adapter for the 30-06 which is interchangeable with the 8 X 57 Mauser. Preceding these aluminum tongs such as you have were the steel tongs which were stamped for the individual cartridge and the shell holes sized for that cartridge. Again, they can be interchangeable (steel tongs) as they often will work with cartridges of the same dimension (base). I don't load 30-06 but I have a set o 30-06 steel tongs that I use to reload my 8mm Mauser cartridges with utilizing a set of 310 8mm Mauser dies.
Yep - a lot of folks pooh pooh the 310 tools and ask "why" when you can use a conventional press and dies, but I batch load my 8mm with a 140 ish grain lead boolit and with my 310 dies set up (lock rings in proper position) I can quickly change the dies out and load just as fast (in my opinion) as I can by using a press - plus, I can "feel" the seating of the boolit and the crimping process. It's just plain fun to see a finished round done with a hand loading tool!
For my 38 specials, I have several sets of the dies that I have set up for various boolit weights/designs - even have a set I can load my 38 Colt Short & Long with. With the proper 310 "neck sizing die", it's an easy operation to neck size my 8mm and 30-30 brass and loading the 30-30 with my 310 dies and tong is a easy and pleasurable process. And loading 45 Colt with the 310 is a hoot. I take my 310 sets with me when I go to AZ and can quickly reload what I shoot up!
As you can tell - I love the 310 tools - but have to admit that I have never tried the Lee Classic Loaders. I have a set for 38 Special as well as 32-20 and once I get my new reloading bench set up (we just moved aver 40 + years) it's on my "to do" list.
Would they look like this balloon head case?
Range pickup brass
Range pickup brass with a 38 Special cartridge