European and American calls cost the same on non-dividend paying stocks (on dividend paying stocks, it might make sense to exercise an American just before the ex-date).
Either way, as was pointed out, in reality BS is used as a deterministic one-to-one mapping between option prices and BS vols. Then, from market quotes (either as prices or as BS vols) a vol-surface is fitted (as a function of strike and expiry time), from which a stochastic process is fitted that correctly re-prices all these points (using a model such as "local vol" or "stochastic vol" or a combination of those two, or others), and then everything is priced of that.
Either way, as was pointed out, in reality BS is used as a deterministic one-to-one mapping between option prices and BS vols. Then, from market quotes (either as prices or as BS vols) a vol-surface is fitted (as a function of strike and expiry time), from which a stochastic process is fitted that correctly re-prices all these points (using a model such as "local vol" or "stochastic vol" or a combination of those two, or others), and then everything is priced of that.