Definition 5.1.1 The random variables $X_{1}, \ldots, X_{n}$ are called a random sample of size $n$ from the population $f(x)$ if $X_{1}, \ldots, X_{n}$ are mutually independent random variables and the marginal pdf or pmf of each $X_{i}$ is the same function $f(x)$. Alternatively, $X_{1}, \ldots, X_{n}$ are called independent and identically distributed random variables with pdf or pmf $f(x)$. This is commonly abbreviated to iid random variables.
What is the difference between these two definitions? Wouldn't both "random sample of size $n$ from the population" and "independent and identically distributed random variables" be the same thing? both have the same density function and are all independent.