What is the major difference between the Telegrapher's Equation and the transmission line equation?
As you can see, the telegrapher's equations are coupled to one another, that is, the voltage equation contains a current term, and the current equation contains a voltage term.

That is why you then see the wave equation, which decouples those (that is, differentiate the telegrapher's voltage equation and plug in your current equation into it) and you end up with:

$$ \dfrac{d^2V(z)}{dz^2}=\gamma ^2V(z)$$
$$ \dfrac{d^2I(z)}{dz^2}=\gamma ^2I(z)$$

With \$\gamma=\sqrt{(R+jwL)(G+jwC)}\$

Those are the differential wave equations.

This is a lot simpler to solve than the telegrapher's equations. Solving those yield exactly what you have for the wave equations.

So what should you use? I'd say most of the time you will use the solutions to the wave equations, because those are the solutions to the telegrapher's equations as well.

What happens is that you start with telegrapher's equations, you then decouple them (differential wave equations) and you finally solve the differential wave equations. The solutions to the differential wave equations is what you probably want to use. All this process just tries to find the solution to telegrapher's equations with the wave equations being an intermediate step along the way.