Stupid question I know.

But what do you guys do to make the difference between your Rhthem and Lead tones, without having them sound like they have no similarity.

Do you just simply raise the volume a tiny bit for the lead, with the same tone? Or do you tweak as much as possible to get a similar, yet different, tone and have that over the rhythem.
Depends whether it's live or in a studio. In the studio you can switch guitars, amps--everything! But live you can't really do that. "Wait a sec. I gotta get a new guitar for this bit of the song! Don't go anywhere." So live, I often change amp channels, pickups, or whatever. Changing amp channels usually gives me a bit more volume, maybe more dirt, and even a different EQ setting.
Rythym sets the rythym, hence the name. The lead holds a melody and just does its own thing as well. The lead should be more confident with the notes he is plays and a clear thinker if solos are improvised.

generally i turn down the treble on rythym and turn it up on lead but it just has to do with the sound you want... you can adjust amp settings at gigs, noone will care.