The idea of a tensor product is to link two Hilbert spaces together in a nice mathematical fashion so that we can work with the combined system. Normally, these two Hilbert spaces each consist of at least one qubit, and sometimes more.
Let's say we have a qubit, which we label $a$, and a qubit which we label $b$. These qubits 'live' in the Hilbert spaces of $\mathcal{H}_{a}$ and $\mathcal{H}_{b}$, respectively; we might call their respective states $|\psi_{a}\rangle$ and $|\psi_{b}\rangle$. The idea of the tensor product is that we can write the state of the two system together as:
$$|\psi_{ab}\rangle = |\psi_{a}\rangle \otimes |\psi_{b}\rangle.$$ We have 'linked' the Hilbert spaces $\mathcal{H}_{a}$ and $\mathcal{H}_{b}$ together into one big composite Hilbert space $\mathcal{H}_{ab}$:
$$
\mathcal{H_{ab}} = \mathcal{H}_{a} \otimes \mathcal{H}_{b}.
$$
Of course, there is no reason that qubit $a$ should come before qubit $b$. We thus also could link their Hilbert spaces together in reversed order:
$$
\mathcal{H_{ba}} = \mathcal{H}_{b} \otimes \mathcal{H}_{a}.
$$
We need to respect our new ordering, and therefore the state of the two systems together is now:
$$
|\psi_{ba}\rangle = |\psi_{b}\rangle \otimes |\psi_{a}\rangle.
$$
Mathematically speaking, this is a different vector than $|\psi_{ab}\rangle$. This is exactly because we have rearranged the order of qubits in how we linked them together.
Explicit example
Let's say that we have a qubit $a$ in the Hilbert space $\mathcal{H}_{a}$ with the state $$|\psi_{a}\rangle = \alpha |0_{a}\rangle + \beta|1_{a}\rangle,$$ and a qubit $b$ in the Hilbert space $\mathcal{H}_{b}$ with the state $$|\psi_{b}\rangle = \gamma |0_{b}\rangle + \delta|1_{b}\rangle.$$
We can link these two qubits together with $a$ first:
$$
|\psi_{ab}\rangle = \alpha\gamma |0_{a}0_{b}\rangle + \alpha\delta |0_{a}1_{b}\rangle + \beta\gamma |1_{a}0_{b}\rangle + \beta\delta|1_{a}1_{b}\rangle,
$$
where I now have specifically labeled the basis states for qubit $a$ and $b$.
Or with $b$ first:
$$
|\psi_{ba}\rangle = \alpha\gamma |0_{b}0_{a}\rangle + \beta\gamma|0_{b}1_{a}\rangle + \alpha\delta |1_{b}0_{a}\rangle + \beta\delta|1_{b}1_{a}\rangle.
$$
These state are not the same. We see that the coefficients for $|01\rangle$ and $|10\rangle$ have been interchanged, but why this happened becomes very obvious if we look at the labels $a$ and $b$ of the basis states. All we have done is writing $a$ or $b$ first.
As an added argument, you could have the SWAP operation act on either of these states, and arrive at the other one. Note that, if we are very scrupulous, strictly speaking, by applying the SWAP gate we have not (re)reversed the order, but we have just 'given' the state of qubit $a$ to qubit $b$ and vice versa. If you may, it is kind of like a 'double fault', that cancels itself out.
So in general a tensor product does not commute, but rearranging the terms is just reordering the systems that you link. We just stick with one particular ordering, and it is always evident which one this is.