Daily driver here - its certainly more harsh than stock, BUT I didn't by my Stang to ride like a Caddy. I'd say the trade off is more than worth it both for looks and handling!
If memory serves me correctly, and I'm a bit rusty as I fell off the Mustang bandwagon for a few years , the dampers are just another word for the replacement shocks/struts that come with the kit. If you're going to lower you're car the right way, you need to replace the shocks/struts too to compensate for the lower ride height - especially since the FR kit drops the car a GOOD amount.
Coil overs place the shocks/struts inside the coil spring itself, rather than having the two located in different areas. Coilover thus refers to the coil spring being "over" the shock/strut; that makes for even better handling than just the FR kit alone (if anyone can add to that or correct any areas I steered right off the track, please do!).
That about sums it up. If you are just looking to lower the car and ultimate performance isn't needed stick with the springs and stock dampner/shocks. Coil overs are adjustable so you can actually change the ride hieght as you see fit...ie for different tracks and such......but, for a good set the cost almost double.