Why is the dynamic programming algorithm of the knapsack problem not polynomial?

When we say polynomial or exponential, we mean polynomial or exponential in some variable. $nW$ is polynomial in $n$ and $W$. However, we usually consider the running time of an algorithm as a function of the size of the input. This is where the argument about $log W$ comes in. Ignoring the values of the items for the moment (and considering only their weights), the input of the knapsack problem is $n$ numbers $leq W$. Each number is represented with $log W$ bits, and there are $n$ numbers, so the size of the instance is $nlog W$. This makes $nW$ not polynomial in the size of the instance, since $W$ is exponential in $log W$. However, (coming back to your example about sorting) an instance for sorting consists of $n$ elements to be sorted, and representing these takes at least $n$ bits (and probably a bit more, since the elements themselves are probably bigger than $1$ bit each). We can represent the number $n$ using $log n$ bits, but we can’t represent $n$ things using $log n$ bits. The main difference is that $W$ represents a number in the input, while $n$ represents “the number of things”. Note that if we were to “cheat”, we could represent $W$ using $W$ bits if we used a unary encoding. This version of the problem is technically polynomial, but really only because we were deliberately being less efficient. It turns out that there is a whole class of Strongly NP-Hard Problems, which are still NP-hard even when the input is represented in unary format. But Knapsack is not one of these problems.