It would take many pages to provide a complete answer to your question. Briefly,
- The H-L estimate is robust to extreme values b/c it is based on robust statistics.
- The CI for H-L is computed based on the asymptotic distribution of the H-L statistic under the null hypothesis (no difference between group locations)
- The bootstrap estimate is based on assuming that the population is well-represented by the sample. If your sample is small, contains rounded values, or contains a disproportional number of unusual observations, it might not be representative.
- In particular, a bootstrap analysis that involves medians of rounded values can lead to situations where the bootstrap distribution is not a good appr... to the actual distribution of the statistic. In those cases, you might need to use the smooth bootstrap.
- The bootstrap process gives you the approximate distribution of the statistic for the sample sizes w/o using any asymptotics.
If you choose to pursue bootstrapping, the answer should be close to the H-L estimates, so you have some basis for deciding whether your bootstrap analysis is correct. As for coding, you can start by reading "The Essential Guide to Bootstrapping in SAS,"
Although your advisor said to use bootstrapping, I think this problem is actually a "permutation test" rather than a bootstrap problem. You can see the article "Resampling and permutation tests in SAS," which computes a permutation test for the mean-difference problem. If you have SAS/IML, you can replace each call to the MEAN function with a call to the MEDIAN function.
