$x^3+x^2+x+1\equiv (x+1)(x^2+1) \pmod 2$, so $x\equiv1$ is the root of multiplicity $3$. Thus we can rewrite the polynomial as $(x+1)^3$. But is this the same as just $x+1$ in $\mathbb{Z}/2\mathbb{Z}[x]$? I'm not so sure, because the multiplicity of the roots is different in the two cases.
A clarification would be appreciated.
There is a distinction that must be made in these types of questions:
Are you talking about the function or the polynomial?
As functions on $\mathbb{Z}/2$, $x+1$ and $(x+1)^3\equiv x^3+x^2+x+1$ are identical. If you take a field extension of $\mathbb{Z}/2$, then you may find that $x+1$ and $(x+1)^3$ are different functions.
However, as polynomials, $x+1$ and $(x+1)^3$ are very different. The difference is that $x+1$ and $(x+1)^3$ have different algebraic properties in the ring $(\mathbb{Z}/2)[x]$. For example $x+1$ is irreducible while $(x+1)^3$ is not irreducible. If you were to take quotients by the ideals that these generate, the quotient by $\langle x+1\rangle$ is a field (it's actually $\mathbb{Z}/2$) while the quotient by $\langle (x+1)^3\rangle$ has zero divisors.
The distinction is: do you treat a polynomial as just the values that arise after plugging in or do you look at the algebraic properties of the polynomial in the ring of polynomials.