Different cabs, or different impulse responses (IR's for short) of those cabs within a plugin, are arguably THE biggest factor in the tone.
Running totally different amps run through the same cab IR will sound more similar than the same amp run through different cab IR's does.
Going direct from either of your real amps removes the cab from the equation entirely.
A bass cab is primarily filtering out a LOT of top end information (some cabs start to roll off high end as low as 2kHz) as well as sub information, and is also introducing some peaks and valleys into the midrange.
Have fun experimenting!