Why would we use a t-distribution instead of the standard normal distribution?

The reason tdistribution is used in inference instead of normal is due to the fact that the theoretical distribution of some estimators is normal (Gaussian) only when the standard deviation is known, and when it is unknown the theoretical distribution is Student t. We rarely know the standard deviation.

Why do we use the t-distribution instead of the standard normal distribution when calculating a confidence interval for a mean how does the t-distribution differ from the standard normal distribution?

The tdistribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the z-distribution).

How is the t-distribution similar to the standard normal distribution?

The T distribution is similar to the normal distribution, just with fatter tails. Both assume a normally distributed population. T distributions have higher kurtosis than normal distributions. The probability of getting values very far from the mean is larger with a T distribution than a normal distribution.