First, efficiency is important here. You can simulate any quantum circuit on a big enough classical computer, it just happens that the size of the required classical computer is exponential in the number of non-Clifford gates in it.
You cannot equate "being universal for" and "can simulate" when considering different computation models.
When you simulate a Clifford circuit with a classical computer, you do not end up building a quantum gate. Instead, you are able to get as much information from the circuit in the classical computation model as you could get from the equivalent quantum circuit (through application to stabilizer states and measurements). Gottesman-Knill essentially tells you that there is little classical advantage to get from Clifford quantum circuits if you have access to a classical computer.
When you decompose (and approximate) an arbitrary quantum gate using {Clifford + Toffoli}, you build a quantum circuit. Your circuit will behave (almost) exactly like the initial gate and this within the same quantum computation model.
Said differently:
"A is universal for B" $\to$ "A is part of B and anything from B can be done with A"
"C can simulate D" $\to$ "From the perspective of C, a black box from D can be emulated with elements from C"
A classical Toffoli gate is universal for reversible computing and in turn for classical computing. A quantum Toffoli gate can simulate a classical Toffoli gate, if given only $|0\rangle$ and $|1\rangle$ inputs.
This means you can build a quantum circuit that will mimic the behaviour of a classical computer only from quantum Toffoli gates, provided you give basis states as inputs.
If you choose this ersatz of classical computer to simulate some Clifford circuits (let's say a Hadamard gate), you can then have access to {quantum Toffoli gates + classical information that can be extracted from said circuit} i.e. some quantum Toffoli gates and the fact that the quantum circuit outputs $|+\rangle$ on $|0\rangle$ and $|-\rangle$ on $|1\rangle$.
This is very different from {quantum Toffoli + Hadamard} because you cannot apply any Hadamard gate, at most you can switch some qubits you know are in a $Z$-basis state to a $X$-basis state. Indeed, the classical simulation of the circuit is bound to take a basis-state input, so you cannot use it for an all-purpose Hadamard gate. Therefore, you cannot make interact the quantum Toffoli with the simulated Hadamard, as the simulation only transcribe the classical behaviour of the Hadamard gate
To sum up, the quantum Toffoli gate:
- can simulate any classical computation (as it can simulate a classical Toffoli gate if given basis-state inputs)
- can (relatively) efficiently simulate the classical behaviour of any Clifford circuit
- can simulate the classical behaviour of any quantum circuit (although very poorly to the best of our knowledge)
- cannot be universal for quantum computing, because it cannot implement a Hadamard gate.
