the taste will be a bit different since brown sugar has stronger flavour than white and is less sweet. but if u want just a coffe it doesn't really matter does it
ANSWER FROM A DIFFERENT EDITOR:
brown sugar is already partly caramelized, standard white sugar is not (or not very much, depending on how precisely you define caramelized, & how pure the sugar is).
in the process of making toffee, you are caramelizing the sugars more, in either case.
the exact result would depend on exactly how you make the toffee & especially on how long/how hot you cook it; but as a general/rule/of thumb, expect brown sugar toffee to taste more "caramely" than white sugar toffee; at least, unless you cook the white sugar more i.e.: longer and/or hotter) than you cook the brown.
if you caramelize the white sugar enough, the difference is extremely minor.
if you use white sugar, & minimize the cooking, or keep the cooking temperature extremely low, the result will not have very much caramel taste, but it will also be harder to produce the other "toffee-like" characteristics; caramelization is one of the defining characteristics of most toffees, and a key part of the process of making them
if you over-cook either type, it burns just the same :P
it is also possible to make toffee from just about any type of sugar, or sugary substance, like honey, or various sugary syrups. pretty much all sugars heat-caramelize.