I can imagine a frequentist density forecast/prediction as something like a distribution of confidence intervals. For instance providing something like the image below with multiple confidence boundary lines (the original is [here][1] with only a single 95% confidence interval). And something similar can be done with prediction intervals.

[![example of a distribution of confidence intervals][2]][2]

With this interpretation the difference between the frequentist density and the Bayesian density corresponds to the difference between a confidence interval and a credible interval. Those two are [not the same][3].

We could say that the Bayesian analysis uses 

 1. More/different information (it includes a posterior distribution for the distribution of parameters, either based on former knowledge or based on assumptions/believes)
 2. Expresses a probability in a different way. The confidence interval relates to the probability of the observed data to occur given the parameters in the range. The credible interval relates to the probability that the parameters occur given the observed data.

The confidence intervals are maybe more easy to interpret than prediction intervals. Prediction intervals include the error of the mean (which can be seen to coincide with confidence intervals) *plus* an estimate of the random noise. 

It is more difficult to give prediction intervals a same frequentist interpretation, although an alternative way to look at is that for frequentist prediction intervals you can say that *'the prediction interval will contain the future observation a fraction $x \%$ of the time'*. So the difference between a frequentist prediction intervals and Bayesian prediction intervals is still that the Bayesian intervals use more information, but the frequentist prediction interval are independent of from the parameter distribution and 'work' independent from the prior distribution (given that the model is correct). I imagine that the following interpretation still works *'the frequentist prediction interval the collection of those predicted values for which the observed effects in the data, or smaller, will occur $x \%$ of the time'*.


  [1]: https://stats.stackexchange.com/a/380218/164061
  [2]: https://i.sstatic.net/wcMJ4.png
  [3]: https://stats.stackexchange.com/a/355164/164061