STZ is more specific to beta cells rather than Alloxan:

STZ seems to be more specific as there is uptake by beta cell specific GLUT2, but not by other glucose transporters. Alloxan not only has toxic effects on islets of Langerhans, but also affect other body organs. It usually produces severe diabetes.

STZ is preferred than Alloxan to induce diabetic animals for its higher inductive rate and lower toxicity. STZ is less toxic than Alloxan. Mortality rate of animals is higher by Alloxan than STZ due to a high loss in body weight.

Please check out the following papers:

Excellent answer Fahimeh mam. One fact I know from personal experience is that the diabetes induced by alloxan, but not STZ, is reversible. I mean, the induction of diabetes is not consistent and is reversed even before introducing pharmacological intervention. 

Go for low dose induction protocol with STZ.

Thank you very much Fahimeh Kazemi madam for your answer.

I perfectly agree with the preceding comments. STZ is preferable.