Why use a t-distribution instead of a normal distribution for small sample tests? It is commonly said that sample sizes <~30, you are unable to use a z-test for a difference of means due to the CLT. Instead it is said that your sample follows a t-distribution, and that you should use a t-test instead.
The main difference between a normal distribution and t-distribution, as I have heard it described, is that a t-distribution has fatter tails, and is a bit more pointy.
But why use a t-distribution in the first place? If you want fatter tails, you can simply take a normal distribution, and adjust $\sigma$ to be lower. Why is there a need to use a t-distribution for small samples? (It sure looks an awful lot like a normal distribution)

You probably mean "adjust $\sigma$ to be higher". And there's the rub: by how much do you need to adjust your $\sigma$?
Yes, of course you can find a value $\sigma$ such that an $N(0,\sigma^2)$ distribution has the same critical value, or will give the same $p$ value as a $t$ distribution with $k$ degrees of freedom. But your $\sigma$ will depend on $k$ (and on whether you only want critical values, or also $p$ values - and if the latter, also on the value of your test statistic).
Conversely, we have a very well thought-through theory and actual mathematical proofs that your test statistics under the null hypothesis are indeed $t$ distributed. So let's actually use that toolset we have.
