Let me contrast two amps in my studio.
First is the Fezzter Prince G., a clone of a tweed Fender Princeton. It has three tubes in it (once 12ax7 for a preamp, one 6L6 for power), and when you crank it, the overdrive you hear is primarily one tube, the 12ax7, being overdriven (power tube overdrive is possible, but you have to dime the amp). This kind of overdrive is crunchy, "hairy," the classic rock and roll OD sound.
The other amp is a Fuchs Overdrive Supreme 50, a Dumble clone. Instead of adding gain all at once at just one point in the circuit, the ODS introduces gain at several points along the circuit. Depending on how you set the controls, you can get a huge amount of gain at the output without ever really blasting any one tube... so the result is a long, singing, smooth sustain with very little "grit." This sound is associated with jazz-blues guitarists like Larry Carlton and Robben Ford.
The difference is in what the amp allows you to do in terms of gain staging.