It's not so much the cable that makes the difference as what it's attached to.
The quarter inch jack inputs on your Firebox are designed for you to plug in an instrument like an electric guitar or perhaps a keyboard output. The output of electric guitars is louder than a microphone and has a much higher impedance than your SM57. This means that, if you plug in your microphone where an instrument is designed to go, the mic will be quieter than it's supposed to be and, because of the impedance issue, the frequency response will be slightly different than the manufacturer planned.
There's one other difference too. Using an XLR to XLR will give you a "balanced" input to your Firebox which is very effective at reducing electrical interference on your cable. Using the TS (what you called "mono") connector gives you an unbalanced circuit which is more susceptible to electrical interference. Over a short run this probably won't be too much of an issue but it could be significant if you have a sparking thermostat or something in your house. And, because you'll have to crank up the input gain when you plug a mic into an instrument input, any noise will be amplified at that stage.
So, although the cable itself doesn't make too much difference, using the wrong input connector can.
Bob