Given $n \in \mathbb{N}$, there exists an algebra $A_n$ such that $\text{Findim}(A_n) = n - 1$ but $r.\text{gl.dim}(A_n) = \infty$.
Consider the following quiver algebra $A_n$:
with the relation that the composition of any two arrows is zero, i.e., $r^2 = 0$.
Clearly, $A_n$ is a monomial algebra. According to the paper Finitistic dimensions of finite dimensional monomial algebras, it follows that $\text{Findim}(A_n) = n-1$. Moreover, it is straightforward to compute that $r.\text{gl.dim}(A_n) = \infty$.
Therefore, we have $\text{Findim}(A_n) = n-1 < \infty = r.\text{gl.dim}(A_n)$ for $n \geq 1$.