A big part of drive for a blues player is being able to control the dynamics. On a natural tube amp, you can turn up to just past the point that it really starts giving it - with your guitar volumes up full. Turn it up more, and all that really happens is that it starts to compress more, and you have less control over the dynamics.
Once you're at that point, you can control the amount of drive by how hard you pick, and with your guitar volumes. This, IMO, is really essential for blues tone. It doesn't matter what tone you prefer, you still want to have the dynamics available to you.
I have a 4x10 bassman. But it's a '70s silverface model. It barely overdrives no matter how loud I crank it. And this amp is loud. Waaaay too loud for any practical application I have. My solution to live playing is a tubescreamer. Very warm overdrive and singing sustain, but limited dynamics. It cleans up a little if I pick very lightly. But it's not what you'd call responsive. So I'm looking at getting this:
It'll put all the dynamics back in a Fender voiced package. Best of all, it'll do it at any volume.
But for recording, I really think that Alan's hit it on the nail. For practical purposes, a modern amp modeller is the best way to go. You can create mic combinations you couldn't afford, play at volumes you'd never be able to, and use classic, vintage amp models that you'd never be able to find or afford made out of tweed, timber, and tubes.
The latest generation are at such high resolution, and have such nuanced dynamics, that you'll almost certainly get a better sound on "tape" than you would as a newbie to amp mic'ing without your own studio.
IK Multimedia's Amplitube Fender is amazing. The playing dynamics are fantastic. The amps models stack up great side-by-side with the real things.