From my perspective there is zero conflict and in fact the two work very well together. However, which one is better for you?
Pilates is certainly more focused on strenghtening your core than Yoga and I have seen yoga students progress rapidly in poses such as Tolasana or Pendulum pose when they have started to do real core strengthening work in Pilates class.
Likewise, clients who have been doing Pilates for a while and have reaped the benefits of increased joint mobility, strengthened and toned mid-section and core but still remain relatively tight in hamstrings and hips, will gain flexibility much more rapidly in a Yoga class.
The question isn't which is better, but which is better for you.
If you are unsure as to which class is better for you just send a quick email to email@example.com or contact me at www.facebook.com/therapy.yoga and I will be happy to help you choose the class or combination of classes that is right for you.