There is definitely a difference between “stick to” and “stick with” in general. When you add “the plan” they become more similar but in my usage some differences are retained. “Stick with the plan” is softer, like saying “Continue doing what we planned”. “Stick to the plan” is like saying “Don’t deviate from what we planned.” Continuity vs conformity.

I think that there is a difference, and I disagree with the idea that it depends on where you come from. I speak BrE, but DavidK's AmE perspective and explanation makes perfect sense to me. 'Stick with [something]' is reassuring and supportive - it means don't give up. By contrast 'stick to [something]' implies strict adherence. Upvote for DavidK.

I would agree with Su Ki and DavidK. I think it is possible to stick with a plan but not to stick to it if the plan is not totally abandoned but mostly followed with a few minor deviations. I would tend to say to somebody 'stick with the plan' if I thought they were likely to totally abandon it but 'stick to...' if I thought they were going to just deviate

No discernible difference to me. If there is one, it's not worth worrying about (except for the pedants - that's what they do).

Despite the thread on Stackexchange, with which I disagree, the answer may lie in the different countries that speak English. I have often heard American users say 'stick with', but not really in the UK or in South Africa. We always say 'stick to.' Stackexchange may reference American usages, but not, as far as I am aware, British or South African.