I would say that the 2 main differences between ANNOVAR and SeattleSeq would be the following:
ANNOVAR is a command line tool wich needs you to locally download the annotation tables of your interest before being able to annotate, while SeattleSeq is a web tool that only requires you to send the list of variants and liberates you from computer power needed to annotate (though you'll depend on the system batch queue to get your results)
SeattleSeq offers an fine yet fixed amount of annotations which may or may not be enough for your interesets, while ANNOVAR allows you to use many more annotations (not only the ones described on its website, but any UCSC table).
although there are other annotation tools like snpEff or Ensembl's Variant Effect Predictor which can be considered as faster yet limited alternatives, I also ended up having to make a decision among the previous ones. I needed the annotation process to be very flexible because the experiments at our lab where not going to be very similar among them, and I wanted to be able to customize the annotation output depending on what a particular researcher would require. I also wanted to have control of the whole process, and depending on an external computing resource (I don't mind using web resources, but only when they have added values like frequent updates, which is not the case of SeattleSeq) when we could implement it locally very easily didn't make sense for me. finally, the fact that SeattleSeq didn't have annotations that were demanded by all the researchers that were contacting me (such as 1000 genomes frequency or ensembl genes) finally tipped the balance in favor of ANNOVAR.
depending on your particular annotation interest, you may use this rationale to decide whether to use one or the other. either way, I wouldn't consider using both.
It looks like both provide similar annotation of variants (GERP, SIFT scores etc). Annovar may provide a little more annotation about whether variants are present in areas with linkage to clinical phenotypes in GWAS data. One key difference is Seattleseq uses a web-based interface, while Annovar can be run from the command-line, allowing it potentially to be integrated into Galaxy workflows.