ANOM compares the group mean to the overall mean, where ANOVA compares the group mean to the other group means. ANOVA can tell you if there is a significant difference, while ANOM can tell which group mean is significantly different.
The SAS documentation on PROC ANOM has better discussion than I can provide:
http://support.sas.com/publishing/pubcat/chaps/59050.pdf
Regards,
Mark