I wrote an article on the main TDPRI page about this a little while back.
As others have mentioned, negative feedback widens the frequency bandwidth (lower bass, higher treble). It also smooths the frequency response, which means it evens out frequencies across the board (the lows, mids, and treble are all at sorta more equal volumes). Finally, it increases clean headroom. This is not to say your amp will be louder overall. Instead this means overdrive is held at bay for more of the turn on the Volume knobs.
Finally, theoretically, negative feedback is sort of "defeated" or negated when the amp starts clipping (ie distorting). This sort of depends on where in the circuit the NF loop is inserted, the amp, the NF circuit (output stage to preamp or locally like a preamp's cathode bypass), where overdrive occurs (in the preamp for instance), and a few other variables. In almost all cases, in my experience, even when the amp is fully dimed I still hear a difference either with NF or when it's off... mostly in frequency/tone and not in overdrive.
Cathode bypass capacitors are a form of NF and in cases like that it increases preamp gain, therefore overdrive - again, when and where in the circuit overdrive/clipping occurs matters. This might seem counterintuitive to the claims that NF increases headroom. Also, when used in cathode bypass circuits it is usually frequency dependant.
Presence controls are NF with a capacitor to select which frequencies are not fed back, therefore it sound like they are increased (cranking a Presence knob usually sounds more bright and trebley). Resonance controls are the opposite of Presence controls and are NF where bass freqs bypass being fed back and sound more pronounced.
In the end, it matters what you hear. What differences do you notice at different volumes with the NF on or off?