Well, technically, headphone jacks have a small power amplifier to drive them, whereas the send outputs don't - they're referred to as 'line-level' outputs and are meant to connect to devices which don't need a lot of power, such as amplifier inputs or effect units. You can get a sound through headphones but it might not be very loud (and the sound may only come out of one side as the sockets are not wired correctly for stereo headphones). Having said that, I use this trick for checking the presence of line-level signals but wouldn't recommend it for continued use.
Re. quality, some people (including me) have a small amount of hiss on the headphone jack at the front of the machine, but all the outputs on the back of the machine should be high quality. If you were being picky, the shaded area on the right hand side of the block diagram implies the stereo and send outputs are the cleanest, followed by the monitor outputs (which have an extra amplifier stage), followed by the headphones (which have yet another amp stage - probably the cause of the hiss?).