They are different concepts, either you use statistical significance or you do Bayesian updating (ie. using priors):
If you are using a 5% threshold roughly speaking this means that you will accept a hypothesis if the chance of getting equally strong data if your hypothesis is false is 5% or less.
If you are doing Bayesian updating you start with a probability for how likely a statement is (this is your prior) and update based on how likely your data would be if your statement was true or false.
here is an xkcd which highlights the difference: https://xkcd.com/1132/
How does using priors affect the concept of statistical significance? The scientific convention is to use a 5% threshold for significance, no matter whether the hypothesis has been given a low or a high prior probability.
If we momentarily disregard the fact that there might be general methodological issues with using statistical significance, how does the use of priors specifically affect the appropriateness of using statistical significance?