This is a very interesting question. The best answer I can give is that they are used in colocations, and we have customary or traditional choices that are used in different contexts. I am going to give some examples off the top of my head. I am not going to try to reverse-engineer the reasons why we use the words we use.
We speak of "a spool of thread," never a "reel."
The device for winding up fishing line is always a "fishing reel," never a "spool."
Camera film for a still camera--a meter or two of film, wound into a cylinder that is longer than it is wide--is "a spool of film," never a reel.
Movie film, hundreds of feet long, is wound on a large, flat, circular "reel." To the best of my recollection, the film can be "wound on a reel" or "spooled on a reel," but even though the verb "spooled" can be used, the holder is a "reel."
Old-fashioned magnetic recording tape also is wound on a "reel," and the old-fashioned machines are called "reel-to-reel tape recorders."
Here is one special case. The thread for a sewing machine comes on a spool, but before it can be used, some of it has to be wound onto a very small reel which is always called a "bobbin." As far as I know, it is always called a "bobbin," and the word "bobbin" is not used for anything else.
Another word, "coil," refers to rope, hose, cable, or wire, that has been wound into a circular shape with many turns. Using wire as the example, we can say that the wire has been coiled. The coil can be freestanding, or it can be wound around some kind of central holder. That holder can be called a spool, reel, or core. "Coil" can also mean the combination of a spool together with the coil of wire that is wound around it.
P.S. The operation of bringing an airplane jet engine up to full takeoff speed is called "spooling it up."