One can use Thue's 1909 result to show that the Diophantine equation $Ax^3 + By^3 = C$ ($A,B$ not perfect cubes, $C\neq 0$) has finitely many integer solutions $(x,y)$.
But does there exist a simple way to prove this equation (because I think this is a special equation?
I think both of the answers given avoid the modern treatment of Thue equations, which despite being newer is probably simpler in many ways. As my doctoral advisor is a pioneer in this area I feel obligated to explain these ideas.
Let us first emphasize that Thue's theorem, extending to the theorems of Siegel, Dyson, and Roth, is completely ineffective in the height aspect (i.e., it gives no way at all to control the size of solutions) and performs very badly in the count aspect (i.e., it is in principle possible to extract a bound for the number of solutions, but it is terrible). The height of the solutions can be bounded using effective bounds on linear forms of logarithms, due to Baker. However these height bounds are terrible (super-polynomial in the coefficients of the polynomial and the integer $h$ in the equation $F(x,y) = h$) and are of little practical use. This is still the best general method for height bounds of solutions as far as I know.
In terms of bounding the number of solutions, there has been significant progress. The key idea comes from a paper of Bombieri and Schmidt: one can obtain a fairly cheap way to reduce a Thue equation $F(x,y) = h$ into a family of Thue equations $G_a(x,y) = 1$, where $a \in A$ lies in some finite indexing set. The forms $G_a$ have the same degree as $F$. One then obtains a uniform bound, depending only on the degree $d = \deg F = \deg G_a$, for the number of solutions of Thue equations of the shape $G(x,y) = 1$ which is valid for any binary form $G$ of degree $d$ with non-zero discriminant. Thus, one obtains a bound of the shape $C_d \cdot |A|$, where $C_d$ is the bound for the number of solutions to the unit equation $G(x,y) = 1$ for forms of degree $d$.
This approach exploits the fact that for any integer $m$, the set of solutions to the congruence $F(x,y) \equiv 0 \pmod{m}$ lies in a finite number of lattices of $\mathbb{Z}^2$ (that number could be 0, if the congruence is not solvable). This is because $F$ splits into a product of linear forms over an algebraically closed field. One then writes $h = p_1^{a_1} \cdots p_k^{a_k}$, and since $F(x,y) = h$ has a solution the congruence $F(x,y) \equiv 0 \pmod{p_i^{a_i}}$ is solvable for each $1 \leq i \leq k$. This implies that $F$ has a linear factor over $\mathbb{F}_{p_i}$. Each such factor gives rise to a lattice $\Lambda_j(p_i)$. For simplicity, let us assume that $F$ is unramified at $p_i$. Then we may use Hensel's lemma to show that each of the $\Lambda_j(p_i)$'s can be lifted uniquely to a lattice $\Lambda_j(p_i^{a_i})$ whose elements correspond to solutions of the congruence $F(x,y) \equiv 0 \pmod{p_i^{a_i}}$. The ramified case is a bit more delicate, but is handled in full generality by this paper of Cam Stewart.
Thus, for each $p_i^{a_i}$ we obtain a finite number of lattices (at most $d = \deg F$) $\Lambda_j(p_i^{a_i})$ and we apply a transformation sending $\mathbb{Z}^2$ to $\Lambda_j(p_i^{a_i})$, thus obtain forms $F_{\Lambda_j(p_i^{a_i})}(x,y)$. We have thus exchanged our original Thue equation $F(x,y) = h$ for at most $d$ equations of the form
$$\displaystyle F_{\Lambda_j(p_i^{a_i})}(x,y) = h p_i^{-a_i}.$$
We may then repeat this process, and eventually obtain a finite number $A$ (roughly $d^k = d^{\omega(h)}$ many) of lattices and auxiliary forms $G_a$ with $a \leq A$, and the unit equation $G_a(x,y) = 1$. Assuming we can obtain a bound for these latter equations which depends only on $d$, then we will get a bound roughly of the shape $C_d \cdot d^{\omega(h)}$, which is $O_{d,\epsilon}(h^\epsilon)$ for any $\epsilon > 0$.
How are such uniform bounds achieved? The key observation is that binary forms with integer coefficients and non-zero discriminant has discriminant at least one in absolute value. In particular, we are now in a situation where the discriminant is comparably large compared to the integer $h$. This means that we can rely on the Mahler measure of the form (really the corresponding polynomial $F(x,1)$) and the discriminant alone to bound the number of large solutions, using the Thue-Siegel principle, which is an effective (and very simple, i.e., does not require the construction of auxiliary polynomials as in Thue's method) but fairly weak way to push apart large solutions. One then bounds the small solutions suitably, and then optimize the counting by controlling various parameters.
I should mention that Stewart makes a critical observation in his 1991 JAMS paper linked above, where he exploits this principle further by noting that it is really the property that $h$ is small compared to the discriminant of $F$ that matters. Here one observes that as one replaces the forms $F$ with the forms $F_{\Lambda_j(p_i^{a_i})}$ the discriminant actually increases substantially, thus one can often move into a favourable situation where one obtains a finite number of equations of the shape $G_a(x,y) = g$ with $g$ very small (but not necessarily equal to one) compared to the discriminant of $G_a$ without using all of the prime factors of $h$.
This is still essentially the best known result, despite the passage of nearly 30 years (since Stewart's 1991 paper). Indeed, it is conjectured that there should be a uniform bound depending only on the degree $d$ such that the number of solutions to the Thue equation $F(x,y) = h$ is bounded by $O_d(1)$. This is a special case of the uniform boundedness conjecture, which is known to be true assuming the Bombieri-Lang conjecture (this result is due to Caporaso, Harris, and Mazur) Stewart makes an even stronger conjecture in his aforementioned paper.
Indeed, the Bombieri-Schmidt theorem gives unconditionally the following special case of the uniform boundedness conjecture, which I don't believe can be proved any other way: we consider the family $\mathcal{C}(d)$ of affine curves defined by two objects: a binary form $F$ with non-zero discriminant, integer coefficients, and degree $d$, and a prime number $p$ - then the affine curves
$$C(F,p) : = \{F(x,y) = p\}$$
have a uniformly bounded number of integral points (i.e., primitive solutions to the Thue equation $F(x,y) = p$). Indeed, one can get something very explicit... for instance $2800 d^2$ works. This result does not require any input from the Mordell-Weil rank of the Jacobian of the curve, or any other similar quantity (i.e., Minhyong Kim's notion of global Selmer variety).
It really depends on what you mean by the word "simple" in "simple proof". Any proof that your cubic equation always has finitely many integer solutions will imply that if $D\in\mathbb Z$ is not a perfect cube, then for every $\epsilon>0$ there are only finitely many rational numbers $p/q\in\mathbb Q$ satisfying $$ \left| \frac{p}{q}-\sqrt[3]{D} \right| \le \frac{1}{q^{3+\epsilon}}. $$ This is a moderately deep theorem whose known proofs involve either the auxiliary polynomial methods used in Diophantine approximation and transcendence theory or $p$-adic analytic methods (as noted in other comments/answers, these methods that are associated to the names Thue/Siegel/Roth ... and Skolem ...).
In particular, if by "simple proof" you mean something along the lines of $x^2+3y^2=z^2$ has no non-trivial solutions in integers because we can reduce mod $3$, or $y^2=x^3+7$ has no solutions in integers because (with a little bit of algebra) we can reduce it to a question about squares mod $q$ and use quadratic reciprocity, then the answer to your question is no. There are no proofs that I am aware of that use only, say, congruence considerations and cubic reciprocity.
On the other hand, as has been mentioned in other comments and answers, carrying out Thue's argument for $Ax^3+By^3=C$ is somewhat easier than the general case. But I'm not sure it could be considered a "simple proof."
It can be reduced to Mordell equation: $$Y^2 = X^3 + (4ABC)^2$$ with $Y:=4AB(2By^3-C)$ and $X:=-4ABxy$, which was shown by Mordell to have finitely many integer solutions.
ADDED. M. A. Bennett and A. Ghadermarzi (2015) explored the connection between Mordell equations and cubic Thue equations, and computed all solutions for the former with the free terms below $10^7$ by absolute value.
