Yea this is a tricky one. This is how I learned it (coming from someone starting med school next year). Conversion Disorder is when a mental illness manifests itself in some kind of physical capacity so to speak. So someone with major depression may experience total body numbness or paralysis even though there is no underlying physical issue (like nerve damage for example) that is causing their symptoms.
Dissociative Disorder is also caused solely by mental illness, but instead of manifesting itself in a physical symptom like paralysis discussed above, it will have a mental representation. So people "dissociate" from reality eg. have memory blackouts during high stress times or even form multiple personalities to escape mentally from their suffering. So in essence, Conversion Disorder has a "physical" manifestation and Dissociative Disorder has a "mental" manifestation.
I vaguely remember an MCAT practice question being like "A 40 year old man is going through a divorce and has had memory black outs. What disorder is he presenting with?" The answer of course being Dissociative Disorder since it was again a "mental" symptom. I hope this helps. Good luck OP!