Prime counting function

05AB1E, 3 bytes

!fg

This assumes that factorization built-ins are allowed. Try it online!

!    Compute the factorial of the input.
 f   Determine its unique prime factors.
  g  Get the length of the resulting list.

f=lambda n,k=1,p=1:n/k and p%k+f(n,k+1,p*k*k)

Uses the Wilson's Theorem prime generator. The product p tracks (k-1)!^2, and p%k is 1 for primes and 0 for nonprimes.

:pYFn

Try it online!

I wrote a version that had a really cool explanation of how MATL's matrices work:

:YF!s1=1

But it's no longer relevant. Check out the revision history if you want to see it.

New explanation:

:p      % Compute factorial(input)
  YF    % Get the exponenents of prime factorization
    n   % Get the length of the array

Three bytes saved thanks to Dennis's genius solution