Hi Dasha, this is a great question! The difference is very subtle: if you stick TO something, you persevere. For example, I'm sticking to my diet (even though I'm really craving chocolate). Whereas sticking WITH something is simply continuing, with less insinuation of difficulty. For example, she's sticking with her decision. But I can honestly say as a native speaker, that people use these terms interchangeably all the time without realising! Hope this is clear.

Hi Dasha! There is no difference in this context. Alice