Is there any conceptual difference in the prediction interval for simple linear regression between classical and bayesian statistics with non-informative prior?
Yes. In fact there is no such thing as really non-informative prior, and very often those "non-informative" priors lead to results different from the frequentist approach of likelihood maximization.
However, depending of your parameters you can find a prior matching the frequentist approach. Very often you can in fact find priors at least weakly informative (e.g a salary is positive).This will allow you to derive better confidence interval than with the frequentist approach, especially for small datasets.