In general, nonparametric tests tend to be lower in power and have wider confidence intervals. Regarding bias, that will depend on the specific nonparametric test as well as your null hypothesis. Some nonparametric tests are based on the median vs. the mean. Some are based on the shape vs. the mean, etc. In those situations, there could be a potential bias relative to the mean.
I would imagine that if both methods met assumptions and were examining the same parameter construct, they would tend to generally merge given a large enough sample size. Though as Miner mentions it all depends on which statistical tests you are using.
You can get the median from minimizing the absolute deviation (in contrast to least squares to get the mean). But isn't that a parametric method that estimates the parameter theta by minimizing Sum|y-theta| ?
Greta, I agree with the issues you are positing and have thought about them myself in the past. Yes, to me they are both parameters given the general definition of what parameters are. Though, it comes down to the distribution assumptions not being there in non-parameter approaches. So maybe we think the term parametric and non-parametric deal with the estimate, but it is actually dealing with the sampling distribution, etc.