Numerical constants alone don't fully specify the universe. They don't account for the laws that relate numerical constants to each other. What's missing is a sort of structural constant -- a minimal set of laws that describe all physical phenomena.
If the universe were the execution of a program, knowing these numerical constants would be like knowing all the... well... constants, in the source code, but missing all the code that binds them together.
You are viewing into these constants as numbers in memory in programming language. In physics, a constant actually means a law that describes unchanging way of how things interact. Think of these constants not as of simple numbers (1, 0, -1, etc.), but as of constant ways things interact (1 always attract -1, etc).
You're right, constants are only "half" of the equation, so to speak. The other half is the equations themselves. In information theory, the combination of the two is usually referred to as the Minimum Description Length (MDL):
It would be interesting to know what is the Kolmogorov complexity of the complete description of our universe. Or for that matter, what is the Kolmogorov complexity of our current understanding of the universe.
If the universe were the execution of a program, knowing these numerical constants would be like knowing all the... well... constants, in the source code, but missing all the code that binds them together.