Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

You write that there is no obvious contradiction even when taking traces/norms. That is false. There certainly is a contradiction that way. You simply have to take more than one trace! Instead of just taking the trace of $\sqrt[3]{3}$ down to ${\mathbf Q}$, look at the trace of $\sqrt[3]{3}$ and $\sqrt[3]{3}\sqrt[3]{2}$.

We will assume $\sqrt[3]{3}$ is in ${\mathbf Q}(\sqrt[3]{2})$ and get a contradiction. From that containment we get that $\sqrt[3]{6}$ is also in this field, and therefore ${\mathbf Q}(\sqrt[3]{2}) = {\mathbf Q}(\sqrt[3]{3}) = {\mathbf Q}(\sqrt[3]{6})$. Call this common field $K$. Let us assume $$ \sqrt[3]{3} = a + b\sqrt[3]{2} + c\sqrt[3]{4} $$ for some rational $a$, $b$, and $c$. Viewing $K$ as ${\mathbf Q}(\sqrt[3]{3})$, we have ${\rm Tr}_{K/{\mathbf Q}}(\sqrt[3]{3}) = 0$. Viewing $K$ as ${\mathbf Q}(\sqrt[3]{2})$, we have ${\rm Tr}_{K/{\mathbf Q}}(a + b\sqrt[3]{2} + c\sqrt[3]{4}) = 3a$. Thus $3a = 0$, so $a = 0$. Next, multiply the above equation on both sides by $\sqrt[3]{2}$, leading to $$ \sqrt[3]{6} = 2c + b\sqrt[3]{4}. $$ Viewing $K$ as ${\mathbf Q}(\sqrt[3]{6})$, we have ${\rm Tr}_{K/{\mathbf Q}}(\sqrt[3]{6}) = 0$. Viewing $K$ as ${\mathbf Q}(\sqrt[3]{2})$, we have ${\rm Tr}_{K/{\mathbf Q}}(2c + b\sqrt[3]{4}) = 3(2c)= 6c$. Therfore $6c = 0$, so $c = 0$. That makes the above equation $\sqrt[3]{6} = b\sqrt[3]{4}$, and multiplying both sides of this by $\sqrt[3]{2}$ tells us $\sqrt[3]{12} = b \in {\mathbf Q}$, which is false.

The book "Number Fields" by Marcus has an exercise using this idea to show $\sqrt{3} \not\in {\mathbf Q}(\sqrt[4]{2})$. It is exercise 16 of Chapter 2. You should try that exercise to check that you understand this method. (But once you learn more of algebraic number theory you should find that the ramification ideas mentioned by Pete are a more efficient way to see at a glance why $\sqrt[3]{3}$ can't be in ${\mathbf Q}(\sqrt[3]{2})$.)

Edit: Here is a proof using norms. If ${\mathbf Q}(\sqrt[3]{2}) = {\mathbf Q}(\sqrt[3]{3})$, then passing to Galois closures over ${\mathbf Q}$ implies ${\mathbf Q}(\sqrt[3]{2},\omega) = {\mathbf Q}(\sqrt[3]{3},\omega)$. Write this as $F(\sqrt[3]{2}) = F(\sqrt[3]{3})$, where $F = {\mathbf Q}(\omega)$. Then by Kummer theory $2 = 3\gamma^3$ or $2 = 3^2\gamma^3$ for some $\gamma \in F$. Taking norms of both equations from $F$ to $\mathbf Q$, $4 = 9c^3$ or $4 = 81c^3$ for some rational $c$. This is a contradiction.

Another argument: The question is whether $\mathbb Q(2^{1/3})$ and $\mathbb Q(3^{1/3})$ are the same field. But $\mathbb Q(3^{1/3})$ embeds into $\mathbb Q_2$, while $\mathbb Q(2^{1/3})$ does not. Reason? The binomial series for $(1+2)^{1/3}$ is $2$-adically convergent; and $X^3-2$ is Eisenstein over $\mathbb Z_2$, so any root generates a cubic extension of $\mathbb Q_2$.

Yet another argument: look modulo the prime $31$, where we see that $2\equiv4^3$, in fact $x^3-2=(x-4)(x-7)(x-20)$ in $\mathbb F_{31}[x]$, so that $31$ splits completely in $\mathbb Q(2^{1/3})$. On the other hand, $3$ is a primitive root modulo $31$, and in fact $31$ is inert in $\mathbb Q(3^{1/3})$.

I will make use of some basic algebraic number theory.

Let $K_2 = \mathbb{Q}(\sqrt[3]{2})$ and $K_3 = \mathbb{Q}(\sqrt[3]{3})$. As the polynomials $t^3-2$ and $t^3-3$ are both irreducible, both $K_2$ and $K_3$ are cubic number fields. Thus if $\sqrt[3]{3} \in K_2$ we'd have $K_3 = K_2$.

The discriminant of the polynomial $t^3-3$ is $-243 = -3^5$. The discriminant of the number field $K_3 = \mathbb{Q}(\sqrt[3]{3})$ divides this (in fact they are equal but in general this is a more delicate computation), so it is prime to $2$, and thus the prime $2$ does not ramify in $K_3$. On the other hand, the prime $2$ visibly ramifies in $K_2$, so $2$ does divide the discriminant of $K_2$. So $K_2$ and $K_3$ cannot be equal (or even isomorphic).

In some sense we got lucky to "see" that $2$ ramifies in $K_2$. The following gives a more general looking criterion along similar lines.

Variant: Let $K_1 = \mathbb{Q}[t]/(f_1)$ and $K_2 = \mathbb{Q}[t]/(f_2)$ be two number fields. Let $\tilde{d}_1$ be the discriminant of the polynomial $f_1$ and $\tilde{d}_2$ be the discriminant of the polynomial $f_2$. Let $d_1$ and $d_2$ be the discriminants of $K_1$ and $K_2$, and recall that $\tilde{d}_1 = d_1 \cdot a_1^2$ and $\tilde{d}_2 = d_2 \cdot a_2^2$ for some nonzero integers $a_1$ and $a_2$. Thus if there is a prime $p$ such that $\operatorname{ord}_p(d_1)$ is odd and $p \nmid \tilde{d}_2$, then $p \mid d_1$, $p \nmid d_2$ and thus $p$ ramifies in $K_1$ but not in $K_2$. It follows that $K_1$ cannot be embedded in $K_2$.

Unfortunately the variant does not quite apply here since the discriminant of $t^3-2$ is $-108 = -2^2 3^3$, so it was good that we got lucky.