The difference is subtle, and 90% of the time either one will do the job.
With multi-band compression, you first split the incoming signal into bands using steep filters and then compress each band separately before recombining them. With a dynamic equalizer, there are no steep filters unless you want them, and no need to recombine bands. A dynamic EQ can therefore be more transparent because you avoid issues that are inherent in steep filters.
The spectral effect is also different. Dynamic EQ can be more precisely targeted. With a multi-band compressor, all frequencies within a given band are compressed equally. A dynamic EQ can do this too if you want, but alternatively it can also apply the effect to a specific frequency or as a high- or low-shelf.
Dynamic EQs can also be used for expansion as well as compression. If you've got a vocal, for example, that lacks bite, a dynamic equalizer can be used to boost the upper-mids only for those phrases in which they're lacking while leaving other phrases alone that already have enough bite.
As you can probably guess, I'm a fan of dynamic equalizers. Although either will work 90% of the time, I cannot off the top of my head think of any scenario where a multi-band compressor would do a
better job than a dynamic EQ.