The definition of a sufficient statistic is: Let $X_1,...,X_n$ be a random sample from a distribution indexed by a parameter $\theta$. Let $T$ be a statistic. Suppose that, for every $\theta$ and every possible value $t$ of $T$, the conditional joint distribution of $X_1,...,X_n$ given that $T=t$ depends only on $t$ but not on $\theta$. Then, $T$ is a sufficient statistic for parameter $\theta$.
I feel like I know several pieces of the puzzle (like the factorization theorem) to understanding sufficient statistics but do not have the overall theory down.
My main questions are:
1) Why do they say that $T$ is a sufficient statistic for parameter $\theta$? If $\theta$ were the population mean of a normal distribution, say $\mu$, does it mean that anytime we want to find the probability of, say, $X_1,...,X_n$ occurring in a certain way, that we don't need the value of the mean of the population?
2) In real-life, why do we want to use a sufficient statistic? Is seems that just calculating the statistic shouldn't be that much work (such as the sum of X's) so why do we need it?