Pretty much what John Morgan said. I've tried using the same mouthpiece on both instruments and it's an unhappy and frustrating result. If you look in my sig, you'll see the mouthpieces I use on both instruments. But I also (for ME) regard something the size of the Doug Yeo mouthpiece to be way too large. Similarly, a Schilke 60 doesn't work for me across the range. It took me a lot of experimentation with a number of mouthpieces before capitulating to Doug Elliott's.
One piece of advice: Don't try to go "too large" with the bass trombone mouthpiece. This is highly variable and individualistic. And keep in mind that a bass trombone player (even in a lot of "community band parts") is expected to have an upper range very close to that of the 1st tenor (and certainly of the 2nd tenor) while also having a low range extending into the pedal range with the valves. That isn't easy. Many bass trombonists use a mouthpiece smaller than the DE one I have (which I'd say is roughly equivalent to about a 1.5/1.25-ish piece). In fact, I just recently changed to a 112 rim in order to get a little better access to the pedal stuff -- and the result is it's thrown off my higher register (keep getting the wrong partials!) and I'm working on getting that back.
In terms of the sound, I know exactly where you're coming from. It took me the longest time to get what I felt was a good trombone sound from the horn -- instead of a "obviously a tuba/euphonium player on the wrong instrument" sound. I feel that my embouchure on the trombone is different from my embouchure on the euph (and certainly the tuba), but others will disagree.
Gary Merrill
Wessex EEb Bass tuba (DW 3XL or 2XL)
Mack Brass Compensating Euph (DE N106, Euph J, J9 euph)
Amati Oval Euph (DE 104, Euph J, J6 euph)
1924 Buescher 3-valve Eb tuba (with std US receiver), Kelly 25
Schiller American Heritage 7B clone bass trombone (DE LB K/K10/112/14 Lexan, Brass Ark MV50R)
1947 Olds "Standard" trombone (Olds #3)