I'm going general on this, because there are many specific mics that will fall in here and fit the general description. Some omni dynamics (OD) of note--the EV 635a/PL-5, EV RE-50, Beyer M-101, Sennheiser MD-21, and the Shure-built Realistic 1070. A couple of those are still in production or reissued, and there are other offerings by Shure and Audio-Technica that I know of.
Advantages? A couple are obvious. Most will take SPL as well as any other dynamic mic. And of course they have an omni pickup pattern, although there can still be somewhat of a directional tendency at higher frequencies.
A couple of things aren't so obvious. Some of this becomes clearer when drawn into relief with a typical cardioid dynamic (CD) mic. Most CD's will have a presence boost up top as well as the proximity-boost effect--close micing a guitar speaker is pretty effective at getting a simulated Fletcher-Munson effect when doing control-room volume, keeping some of the loud-amp-in-the-room character. An OD doesn't play this game, so it's nice for contrast--lets say you don't want a mid scoop, but want a more midrangey sound for slide guitar. Most OD's have a pretty flat frequency response within their ranges, so you can often get very flat and accurate mids. Also, if the response goes deep enough, bass on an pressure-gradient omni (dynamics and most small-diaphragm condensor omnis) is generally flatter and truer than on a directional mic. Limited band-width with pure mids can be nice for sounding old-timey or intentionally lo- or mid-fi--the 635a can be great for carving out a niche for a sound in a mix.
The other element is that OD's have the omni pattern advantage with less sensitivity than an omni condensor--less reach, less truck-rumbling-by-two-blocks-away, less computer fan, and so forth. That and their SPL handling generally makes them work quite well for difficult percussion like triangles, tambourines, and so forth--if your omni condensor fails the "car-keys test" your OD might save your butt on that type of application.
Definitely a good way to diversify the tool kit--compared to another flavor of the same-old, you can use them differently and do more things when tracking.
--Bear's Gone Fission