Why does changing $X\to iX$ in the definition of the CNOT make an important difference? How to find a ket $|\psi \rangle$ that illustrates how changing X to iX in the definition of the CNOT gate makes an important difference because of what happens when CNOT is applied to $|\psi \rangle$?
The definition of the CNOT gate is:
$$CNOT = |0\rangle\langle 0| \otimes I +  |1\rangle\langle 1| \otimes X $$
What is the physical meaning of this?
 A: Assuming your question is "What $| \psi_i \rangle $ would have different resulting $| \psi_f \rangle $ having undergone a $C-X$ gate vs. a $C-iX$ gate?"
Consider $| \psi_i \rangle = \frac{1}{\sqrt2} \Big( |00 \rangle + |11 \rangle  \Big)$
$$C-X | \psi_i \rangle = \frac{1}{\sqrt2} \Big( |00 \rangle + |10 \rangle  \Big)$$
$$C-iX | \psi_i \rangle = \frac{1}{\sqrt2} \Big( |00 \rangle + i|10 \rangle  \Big)$$
Obviously these two resulting states are different since there is a relative phase. To make the difference more clear you can try to consider measurements in the Y-basis. Just as a note a $C-U$ gate is a "Controlled $U$ gate", and often times you will find $CNOT$ written as $C-X$ to be consistent with $C-H$ (Controlled Hadamard), $C-Z$ (Controlled Z gate), etc.
The Physical meaning of this is that adding a global phase to a single Qubit Gate on a multi Qubit system results in a change in the multi Qubit system. Notice how if we only had a single Qubit, there would be no way of knowing the difference between $X$ and $iX$, because it would be a global phase; however in a 2 Qubit system, that phase shows itself as a relative phase which indeed is measurable.
A: These two gates only differ by a local unitary, namely
$$
\begin{pmatrix}1&0\\ 0&i\end{pmatrix}
$$
on the control qubit. Thus, there is no important difference between them, at least in terms of their entangling power and and non-local properties - they are exactly equivalent in that respect. But of course, it depends what you deem an important difference, which you would have to clarify in your question.
A: Gates $X$ and $iX$ differs in global phase which is $\pi/2$ (since $i = e^{i\frac{\pi}{2}}$). As two states (gates) which differ in global phase only cannot be distinguished, there is no difference between $X$ and $iX$.
However, the difference appears in controlled versions of the gates. Controlled $iX$ is composed of controlled $X$, i.e. CNOT, and controlled global phase which is in this case described by gate $U1(\pi/2) \otimes I$ where
$$
U1=
\begin{pmatrix}
1 & 0 \\ 0 & e^{i\frac{\pi}{2}}
\end{pmatrix}
$$
So $C-iX = \text{CNOT} (U1(\pi/2) \otimes I)$.
