The question is about attitude parameterisation - a rotation vector is one method of describing an attitude; Euler angles are another.
The ultimate description of a rotation is generally considered to be a 3x3 rotation matrix (in 3-space). However, since there are only 3 degrees of freedom in a rotation matrix, then there is considerable redundancy in its representation. Additionally, a rotation matrix does not really provide an intuitive description of what the rotation looks like.
So, what are some ways we can represent the rotation matrix? We can:
- Perform 3 successive rotations, each around a defined axis (Euler Angles); or
- Perform a single rotation about a fixed axis (Rotation vector)
Hence for Euler angles, we can create our rotation matrix as: $\mathbf{R} = \mathbf{R}_x(\phi) \mathbf{R}_y(\theta) \mathbf{R}_z(\psi)$, which consists three successive rotations; a rotation about Z by angle $\psi$, followed by a rotation about the (new) y-axis by $\theta$ followed by rotation about the new x-axis by $\phi$. Classically, these are roll ($\phi$), pitch ($\theta$) and yaw ($\psi$) in the aerospace world.
A rotation vector $\mathbf{\rho}$ consists of a rotation about axis $\frac{\mathbf{\rho}}{\parallel\mathbf{\rho}\parallel}$ by angle $\parallel\mathbf{\rho}\parallel$, except where $\parallel\mathbf{\rho}\parallel = 0$, in which the rotation matrix is simply the identiy matrix. To recover the rotation matrix, the matrix exponential is used: $\mathbf{R} = \operatorname{exp}(\left[\mathbf{\rho}\right]_\times)$ where $\left[\mathbf{\rho}\right]_\times$ is a skew symmetric matrix constructed as $\left[\mathbf{\rho}\right]_\times = \begin{bmatrix} 0 & -\rho_{z} & \rho_{y} \\ \rho_{z} & 0 & -\rho_{x} \\ -\rho_{y} & \rho_{x} & 0\end{bmatrix}$.
Fortunately, there is a shortcut to calculating the matrix exponential called the Rodrigues Rotation Formula.
In short: Euler angles are not the same as a rotation vector (except for very small angles, but that's another story). They cannot be used interchangeably.
There are many many more representations of attitude. Unit Quaternions are common. If you need a cure for insomnia, take a read of this survey paper on attitude representations.