What is the difference between these two formulas?
They are both related to the price of a stock in the black-scholes model. The fact that the second one uses $t$ as a subscript which means it's not a function of $t$ (in this case time), but I don't understand what each formula is used for.
$$S(t)=S(0)e^{(r-\frac{\sigma^2}{2})t+ \sigma W(t)}$$
$$S_t=S_0e^{\mu t+\sigma B(t)}$$
In the Black-Scholes model the stock price process $\{S(t), t \geq 0\}$ is modelled as $$S(t) = S(0)e^{\mu t+ \sigma W(t)}, \qquad (*)$$ where $\mu$ is the return on the stock and $\sigma$ the volatility. In a market absent of arbitrage, any discounted price is a martingale. In particular, the discounted stock price is a martingale, that is by definition $$\mathbb{E}[e^{-rT} S(t)] = S(0).$$ To proceed, substitute $(*)$ into the above equation and solve for $\mu$. You'll find $\mu = r-\sigma^2/2$. Hence, for this choice of $\mu$ there is no arbitrage, which is a necessary assumption to price options.