These are for sure the 2 most commonly used injectables for facial wrinkles.  Botox has a greater market share and many times the brand name is used broadly even when referring to any of the other 2 available neurotoxins (e.g. Dysport, Xeomin).  From a medical standpoint, their efficacy is comparable.  In general terms, patient's usually have Botox when they are introduced to neurotoxins for the first time. and this could be based on doctor's preference or patient request. There is a small subset of patients that report less than desirable outcomes with Botox, which then prompts them to use Dysport and in many cases they do see a better response.  The same can happen vice versa.  Dysport contain cow's milk protein and lactose. Therefore, someone with allergies to these ingredients might be a better candidate for Botox.
In my practice, I would say approximately 90% of my patients are treated with Botox, 9% with Dysport, and less than 1% with Xeomin.  The patients treated with Xeomin usually have done it before at another practice and they come to me requesting this particular product.  For a first-time user, I tend to use Botox as my treatment of choice.
It is important to mention, that the price point for Dysport could be at times cheaper than Botox and more affordable for patients.  Considering the efficacy is comparable, many patients may elect to have Dysport just for financial reasons.