An imupulse response, is the output you get when you apply an impulse, like a delta dirac function, to your system (only for LTI?).
By knowing the impulse response you know the system.
The transferfunction relates the input to the output. I.e. this is a representation of the system.
So aren't both the same? Or Did I misunderstand something?
Best Answer
For any partially continuous function $f : \mathbb{R} \to \mathbb{R}$, the Dirac delta function has the nice property
$$ f(t) = \int_{-\infty}^\infty f(s) \delta(t - s) ds $$
So, any partially continuous function can be written as a sum of Dirac delta functions. This is particularly useful for LTI systems. If we know the impulse response of a LTI system, we can calculate its output for a specific input function using the above property. In fact, it is called the "convolution integral".
The Laplace transform of the inpulse response is called the transfer function. It is also useful since $\mathcal{L}\{\delta(t)\} = 1$ and $\mathcal{L}\{f * g\} = \mathcal{L}\{f\} \mathcal{L}\{g\}$. Because of this property, it gives a nice rational polynomial representation of input/output behavior of LTI systems.
You may also calculate the impulse response of a time-varying and/or non-linear system, but it isn't useful since you can't use it to calculate the output for any input function. The system must be LTI to do so.