John, Marc's right, same thing.
it is a curiosity that they would be labeled as 3/16 24. Of course it is possible to spec, or even encounter a thread of any pitch and diam, however it seems inconceivable that a home improvement store would be selling a specialty, non standard tap 2 1/2 thou away from one of the most common sizes. If the tap was from some specialty shop who should know tooling, like optics or gun smithing or something I'd be scratching my head a bit, in the case I'd bet Lowes, not a tooling supplier, has had a supplier incorrectly lable the package. typically when you need to thread something 10/24, it's 3/16 stock you grab.
you mention fit as concern. the fit of the thread has nothing to do with how close the OD is to the nominal dia of the thread. For example one could make a 10 24 thread and fit like a micrometer thread starting with .180 stock. It is the relative depths of the grooves that you cut that determines the fit of the threads. Same way as our tap drill sizes typically create a 75% thread engagement but do not produce sloppily fitting threads.
so far as thread fit goes, there are lots of tedious detailed standards you can read up on, but basically the tap and dies have to cut such that there is bit of clearance between male and female thread else things will interfere. the tap should cut a little over nominal and the die a little under. The smaller tolerance you give on this clearance, the tighter the fit feels AND the trickier it is to manufacture in volume. Therefor there is going to be a lot of slop with any run of the mill commercial fastener. This doesn't matter in most applications however sometimes the thread is not for a fixed assembly and we do want a nice fit.
To get that nice smooth thread fit, there are two basic choices. not much we can do with the female threads, tap away. on the male threads either cut them on the lathe to suit the tapped hole - finish with very light cuts trying it in the female threads (or if you want to get fancy make a thread gauge to suit the female thread and measure the thread being cut with wires & micrometer) OR use a split die. A split die has a small screw allowing you to adjust its depth of cut. quality dies are split imo and they should be chosen over solid dies; many times its just the thing to get an otherwise difficult fit working.
