How to discretize a continuous time domain state space model?

Question

Let's say I have a state space model of a linear dynamic system in the continuous time domain in following form

$$\dot{\mathbf{x}} = \mathbf{A}\cdot\mathbf{x} + \mathbf{B}\cdot\mathbf{u}$$

My goal is to implement the above given system in a control software which runs in a real time in a digital signal processor. This piece of software will be used as an open loop state observer of a real dynamic system. For software implementation purposes it is necessary to discretize the above given state space model.

I have a question regarding the discretization process. I have encountered basically two approaches to the discretization process

1. the discretization is done directly via below given transformation of the matrices of the state space model in the continuous time domain based on given sampling period $T$

   \begin{eqnarray} \mathbf{A}_d &=& \mathbf{I} + \frac{\mathbf{A}\cdot T}{2!} + \frac{(\mathbf{A}\cdot T)^2}{3!} + \ldots = \mathbf{I} + \mathbf{A}\cdot T\cdot\mathbf{\Psi} \\ \mathbf{B}_d &=& \mathbf{B}\cdot T\cdot\mathbf{\Psi} \end{eqnarray}

2. the discretization is done indirectly via selection of the numerical method for solving the ordinary differential equations (Euler, Adams-Bashforth, Runge-Kutta etc.)

My question is whether there is any difference (as far as the outcomes of the resulting discrete model) between the above mentioned methods?

Edit:

As far as the first method. The derivation can be found e.g. in Digital Control of Dynamic Systems (Franklin, Powel, Workman) on page 101 - 106.