Are comets and meteors on a continuum?
That is, is the difference just a matter of degree of what temperature their constituents melt at?
Is there a continuous range from ice to “dirty snowball” to “snowy dirtball” to dirtball?
Yes, comets and asteroids are on a continuum.
They go from this rocky body Eros
To rocky bodies that occasionally sprout tails (P/2004 TU12)
To dim, bowling pin shaped comets (Borrelly)
To full on, no foolin comets (Wild)
-That last site has some beautiful images, and a movie of the flyby.
Makes you wonder why aren’t they called the same thing?
Usually science terminology is intent to put new classification to everything you learned in school 
Because they’re not, functionally. Just like cows and sirloin steaks are on a continuum, but aren’t called the same thing.
A comet is an object in an orbit that periodically brings it close enough to the sun to induce a physical change in the components that make it up, causing a vapor/dust tail to form, then eventually going on its way back to the outer part of its orbit and settling back down to big, dirty snowball status.
A meteor (from an Earth perspective) is an object in the same (or a near) orbit as the Earth, which occasionally gets swept into the atmosphere as the planet passes along that path and burns up from friction, becoming a meteorite. Meteors can be the remnants of comets that have “worn out”, but aren’t necessarily.
They’re both rocks in space, but their functions are different. So they’re not the same thing.
Astronomy’s never been very good with the whole classification thing. Witness the debate over whether Pluto is a “planet” or not. It’s all just a taxonomy problem - there’s no good definition of what a planet actually is. Then there’s “planetary nebulae,” which have nothing to do with planets, but they look big and round like planets in small backyard telescopes, and the name stuck. And is a brown dwarf a big planet or a small star?
In the end, it’s a whole bunch of big (or small) lumps of stuff, and Nature doesn’t care what we call it! 
I was under the impression comets were mostly made of light elements frozen into ice, while meteroids were mostly of rocky materials or iron.
That’s the usual way they’re thought of, but when an apparent asteroid develops a tail on close approach to the sun (see my second link), they reclassify it as a comet. It may be 90% rock and 10% ice, but if it grows a tail, it’s a comet.
From my memory of Astronomy 101 Meteor Showers are remnants of comets that have broken up, and formed a band of material following the comet’s old orbit - so they should have roughly the same composition as a comet, but meteors in and of themselves can be anything that enters the earth’s atmosphere. Correct me if I’m wrong.