First off, the terms do not inherent reflect a driver's ideal application.
Terms like "Woofer", "Mid-Woofer", and "Midrange" are really just guideline words used to differentiate one driver from another. One company's "woofer" could very well be another company's "midrange".
That said -
In general, woofers are intended specifically to not have any crossover applied to the bottom end of their operating range. That doesn't every driver referred to as a woofer can reproduce 40hz with power, but it does mean their power handling and input operating range can be well below 40Hz without destroying the woofer during use.
In order to get those characteristics, woofers tend to have larger voice coils, longer peak to peak excursion capabilities, and looser suspensions. This makes them capable of handling lots of bass energy without harsh distortion, but it also reduces their finer capabilities at the top end of their spectrum.
A midrange and most mid-woofers do not have the ability to absorb tons of bass energy without harsh THD or potentially failing. They tend to have short peak to peak excursion limits, potentially underhung voice coils (limiting their throw), and lower power handling. They also tend to have tighter suspensions.
The difference in sound can be heard in the midrange detail, differences in THD characteristics, and "resolution". A good midrange driver will have very low THD in the 500Hz to 5kHz range, even at high SPLs while a good woofer will have a lower THD below 400Hz. The inverse of that is that a good midrange could have higher THD below 400Hz and a good woofer will have higher THD above 500Hz.
Then you could get into the electrical issues that impact performance, things like reactance at different frequencies, the Fs, and other qualities make one more suited for the most realistic midrange performance at that price point versus the most realistic bass peformance for the size and price.
When the SPLs are over 90dB, then you need to consider bass causing THD in the midrange as well as doppler distortion. A true midrange with a crossover at or above 200Hz will inherently have lower THD and doppler distortion in its operating range than an equivilent woofer with no crossover at or around 200Hz. Attemping to move a cone around at 60Hz at real world listening levels just gets in the way of that very same cone attempting to produce very realistic sound in the 1kHz to 3kHz range.
So, my advice is all about getting the cleanest output at real world listening levels, improving peak performance, and reducing all forms of distortion.