Are there any significant differences in our genome compared to the genes of our ancestors from 1000-2000 years ago?
And if there are significant differences, do they result in significant differences in phenotypic traits?
The larger differences are most likely in epigenetic marks on the DNA. The environment is a lot different today than it was 2000 years ago and those differences are stronger determinants of epigenetic change than sequence change.
2000 years is only about 40 generations and that is not very much to see great differences in DNA sequence or allele frequencies - once founder populations are removed from the analysis.
Height is highly influenced by genetics, but can be masked by environment - notably diet. Diets higher in protein content are more likely to elicit the expression of the "height alleles" but only to a point as too much protein is also unhealthy.
Added in Edit 4 Apr 2012: I should add that my response is written from the perspective of frequency of alleles or epigenetic marks across a population.
In the literature, there are examples of DNA analysis from very old specimen. For instance, look at the analysis of both nuclear and mitochondrial DNA from 62 buried humans exhumed from a Mongolian cemetery dating from 2,000 years ago, as described by Keyser-Traqui et al. 2003, Am J Hum Genet 73(2): 247–260. Despite, the publication is open-access, the DNA sequences are not available, and no comparison with current mongolian DNA has been described in the study, but for sure your question should be answerable.