Doukutsu Randamu is the first thing that pops in my mind. Calling it a Cave Story mod feels a bit disingenuous, though, as it's basically a whole dang roguelike using the Cave Story engine as a base. The levels are randomly generated, and there are a ton of items and characters available to use.
Does Not Commute is a pretty funny, short (I think) mod where you have to guide a variety of Cave Story characters to their destinations, all the while avoiding the movements of the characters that you previously controlled. I promise, that explanation will make a lot more sense once you get into the meat of the mod.
Sunset Challenge is a difficult mod in the vein of the CS+ challenges, but also a very pretty mod. Again, short and sweet.
If you're looking for an adventure that takes place before or after the plot of Cave Story, I'd say Evergreen (a prequel) and Another Cave Story Sequel will satisfy in that regard. Both are a little more story-heavy than the original Cave Story, but they also have some interesting additions in the gameplay department that help give them their own identity.
The "This Mod Makes No Sense" Trilogy (Original, Jack's Bogus Adventure, and Twice upon Itoh) are filled with completely bizarre gimmicks that are pretty entertaining to experience, I think. Just beware: this mod has EXTREMELY dumb humor. Like, barely beneath Cavern Tale levels of dumb. That may be a turn-off or a turn-on, depending on your sense of humor. (I am one of those people)
Test your Cave Story know-how in The 2000 IQ Cave Story Quiz! At least... if you think you're ready. Truth be told, some of the questions can get quite devious...
You'll notice I haven't mentioned a lot of older mods, and that's because, truth be told, most of the ones I'm aware of aren't... that good? WTF Story starts out kind of strong, but devolves into weird gimmicks and fetch quests by the end, and I'm sure you've heard the horror stories about Jenka's Nightmare's unfair puzzles already. The only "old" Cave Story mod I'd wholeheartedly recommend is Rave Story, which I feel has aged gracefully compared to others of its ilk.
Aaaaaaand once you're done with the whole list I've given you, don't forget to check out Doukutsu Club for all of your Cave Story modding needs!
First off, you should definitely look at Studio Pixel's other games, especially Kero Blaster. You should find a complete list of the other games here.
As for mods, if you're looking for more of the Cave Story experience, try A Lost Land - this is a pretty old fan-made Cave Story sequel. Unlike a lot of newer mods that innovate on the gameplay, this one here is quite straightforward and similar to the original game (albeit slightly harder). All in all it's a finished, fairly long mod that's fun to play.
Jenka's Nightmare can be a little divisive. The puzzles are quite challenging, but the author includes a very comprehensive walkthrough where you can just look up the answers if you're ever stuck, which is nice. A lot of interesting elements are introduced, the characters are fleshed out more, and the story is well-made. Personally I found it to be overall a very solid experience, on par with or even surpassing the original game (blasphemous, I know), but don't take my word for it - have a play at it yourself!
Coming to mods with a story unrelated to the original game, I'd like to recommend Ordeal Pillar. Very intricate storytelling, and very varied gameplay challenges that should keep you on your toes. The same author also has a couple other mods I really liked, Shipwrecked and Inner Demons.
For more atmospheric experiences, try out Cascade and Lost Machine. Hat Thief is pretty fun and funny, if a bit difficult.
I haven't finished Treasure Islands, but it's pretty interesting. Feels like a different game.
It's been a while, there's more I don't remember, and I wish I could write better descriptions for most of these but whatever. I hope you have fun with the recommendations from this thread, and maybe even make your own mod someday...?
(If you're open to other games that someone who liked Cave Story might also like, then as a very rough guide off the top of my head I'd point you to Vernal Edge, Momodora, Iconoclasts, Owlboy, Catmaze, Phoenotopia, Shovel Knight, Rabi-Ribi, Flynn: Son of Crimson, Transiruby, etc.)