What is the difference between the Euler and Tustin method, when going from time to z- domain?
Tustins approximation is the more accurate of the two in general.  The simple explanation is that Tustin's method is actually mathematically equivalent to trapezoidal integration, while Euler's method (or more generally, any first order difference approximation) uses the values of the ODE at only one discrete measurement point to estimate the value one timestep later.  Because Tustin's method is mathematically equivalent to trapezoidal integration (and therefore equivalent to the "euler midpoint" method or "improved euler method) it generally has lower error because it benefits from the averaging of two discrete points of measurement.  A more sophisticated approach would be to use a "zero order hold" to convert to the Z domain, but in most of my controls work Tustin's method actually outperforms the Zero Order Hold because of non-ideal sampling rates and calculation times.