The EGDM and MH are two of my favorite drives, so I will take a stab, but YYMV....
I used to run both at 12 Volts, but with single coils I found the EGDM was more versatile at 9 Volts. Being my first BJFE I thought I knew it well though it turns out I was just in a rut. I am now rediscovering the 12 - 5 o'clock side of the Voice knob. The EGDM is more versatile than the MHvT I have (I have owned several MHv1s and tried even more). You can run the EGDM from 8 to 18 Volts and it sounds surprisingly different at 9, 12 and 18 V that I have tried it. It works great into a clean or distorted amp and has so much volume on tap (unity is about 8 o'clock) that it can do a dirty volume boost for solos, get lots of distortion and will drive certain dirt pedals in a yummy way. Favorites are EGDM > HBOD, EGDM>MH, EGDM > DRD and EGDM > MR. The EGDM is really picky about what runs into it though a good germanium or silicon Treble Booster or the SYODv2 work really well.
I don't know which version of MH the Bearfoot MH and MH4k are based on as the BF MH4k I have here from the tour box sounds really good, but not exactly the same as any MHv1 or the MHvT (check the library for descriptions) I have tried. The Distortion on the MH is smoother than the slightly fuzzy EGDM and the MH is more responsive to picking dynamics. What is strange about the MH series is even though it has such great picking dynamics it has a great "keraaang!" when laying into some Townsend power chords at volume. The BJFE MHv1 really needs to be played somewhat loud for it to sound good, but the MHvT works well at all volumes. The MH, for me, has few stacking partners as it is so thick already, but the MH > HBOD stack is amazing!
I don't think the EGDM and MH sound very much alike though I think you can tell they were designed by the same person!
I think the reason for the jump in EGDM prices might be threefold. There are about half as many EGDMs as DRDs or HBODs out there and there were a few months where no EGDMs had come up for sale, so I think demand is up due to many more people trying Bjorn's designs via Bearfoot and lower Mad Professor pricing. Since Bearfoot does not make a version of the EGDM the many happy Bearfoot customers are venturing out and trying to find Bjorn's other designs which seems to have kept the used BJFE market from sinking while the economy is down. Lastly, there was a rumor that Bjorn is running out of EGDM parts which has scared a few people into buying now before an official OOP tag is put on the EGDM and prices really skyrocket! I do not know if the part shortage is true, or if it is possible to find additional quality parts, so I guess until supply far out paces demand prices may stay above $500.
For me? I am not selling the two I have as they are so special on several fronts. One was my first BJFE and the other .....
Gunbarrel Guitars, Evidence Audio wires.