In most references from dynamic system theory, the following linear continuous dynamic system is considered. $$\frac{\text{d}x(t)}{\text{d}t}=Ax(t)+Bu(t)+Dd_{1}(t)\quad (1)$$ $$y(t)=Cx(t)+Ed_{2}(t) \quad (2)$$ where $x\in \mathbb{R}^{n}, y\in {{\mathbb{R}}^{p}},d_{1}\in {{\mathbb{R}}^{m}},d_{2}\in{{\mathbb{R}}^{q}}$ represent the state vector, measurement output vector, process disturbance and measurement disturbance vector respectively. $A, B, C, D, E$ are constants matrices of appropriate dimension.
Again, the following linear discrete dynamic system is mostly studied in references. $$x(k+1)=Ax(k)+Bu(k)+Dw_{1}(k)\quad (3)$$ $$y(k)=Cx(k)+Ew_{2}(k)\quad (4)$$ where $x\in \mathbb{R}^{n}, y\in {{\mathbb{R}}^{p}},w_{1}\in {{\mathbb{R}}^{m}},w_{2}\in{{\mathbb{R}}^{q}}$ represent the state vector, measurement output vector, process noise and measurement noise vector respectively.
The questions are stated as following.
1) Are the disturbance $d$ and noise $w$ the same thing? If not, why in continuous system, only disturbance is considered, and only noise is considered in discrete system?
2) In the continuous system, when the disturbance $d$ is stated as a certain function, can the disturbance $d$ be assumed to be differential? Is this assumption reasonable?
3) In the continuous system, when the disturbance $d$ can be stated as a stochastic process such as Gauss white noise, can the disturbance $d$ be assumed to be differential? Is this assumption reasonable?
1 - Yes, same thing, mutatis mutandis.
2 - Reasonable to assume $d$ differentiable, or at least continuous piecewise. But it's also reasonable to think of $d$ as a stochastic process, in which case the math becomes more complicated. Start with the simpler cases, and then proceed to more general models as needed.
3 - No, a Wiener process, of which white noise is a generalized derivative, is nowhere differentiable. Go into these more complex models if needed; otherwise stick with the more restrictive assumptions which make everything easier.