In my opinion they have different indications and sometimes overlapping.
In children postop there is a dip in lv function that can be managed with norepi. Some centers use norepi routinely. On the other side, for example, if you have a patient on amiodarone and becomes hypotensive, vasopressin is probably a better choice to provide vasoconstriction without aggravating the primary arrhythmia.
As Dres Donatelli and Neema have already indicated norepinephrine has several advantages. In our experience vasopressin is added if there is vasoplegia, e.g. in endocarditic patients. In case of high output failure (with preserved LV function) we will give methyleneblue to get vasoreactivity. It is not advisable to use it in patients with poor LV function.
The answer is quite simple. Norepinephrine is the first line vasoactive drug in most centers. However, quite an amount of patients do not respond to increasing doses when suffering from vasoplegia. Here vasopressin at an intial dose of 0.5 to 1.0 I.U/h might revert vasoplegia. As Dr. Aleksic mentioned, in case of severe vasoplegia in combination with preserved LV-Fx methylene blue administered at 1.0 to a max. of 3.0 mg/Kg body weight might be a reasonable alternative. Here I recommend starting at a low dose (0.5-1 mg/kg) and a careful titration until patient response. Avoid overexceeding the maximum dose to avoid toxicity. Before switching to second line vasoactive drugs, care has to be taken, that both volume status and pH has been normalized. Swan-Ganz catheterization is absolutely mandatory in this setup.
Vasopressin does not affect pulmonary vascular resistance and is of benefit when RV impairment is a concern. However as already discussed Norepinephrine is first-line in most instances.