Botox vs Azzalure?
The market leaders for botulinum toxins have been Allergan with Botox, and Ipsen with Dysport. Botox has been so dominant that the use of botulinum toxin has become known as Botox treatment.
Galderma has now brought a smaller vial of the same botulinum toxin as Dysport to the market with the trade name of Azzalure.
The manufacturers obviously want to persuade us that their product is the better product. The 
reality, in my opinion, is that Botox, Dysport and Azzalure are all equally good. There is a subtle difference in that at the same volume of injection Dysport and Azzalure spread a little further. This is useful when treating the forehead but less useful closer to the eye. The solution when using these preparations is to use higher dose in smaller volume around the eye.
In summary, I wouldn't worry which botulinum your practioner uses so long as they are familar with its characteristics.
It is worth knowing which product you are being given, and at what dose, such that if you move to a different practitioner they know what your preferences are.