Difference between $\mathcal F_\tau$ and $\sigma(\tau)$

As you say, $\mathcal{F}_\tau$ contains all the information up to the stopping time, while $\sigma(\tau)$ only contains information that is relevant to the stopping.

For example, take $X_1,X_2,\ldots,X_N$ iid with continuous distribution, $\tau=\inf\{i:X_{i-1}>0,X_i>0\}\wedge N$, and $S_k=\sum_{i=1}^{k\wedge \tau}X_i$. That is, a player plays the same game until he gets a profit twice in a row. Then the total result of the game, $S_N$, actually equals $S_\tau$, and so it is $\mathcal{F}_\tau$-measurable. However, it is far from $\sigma(\tau)$-measurable, in fact knowing the value of $\tau$ doesn't even tell you the sign of $S_N$, that is, whether the player made a total profit or loss (unless of course $\tau=2$).