A whole genome scan can add more insight. 23andme tests about 10% of known SNPs (~300k SNPs out of 3m). If you have a novel SNP or important insertion/deletions, that would not be detected by 23andme either. Enlis is the best software for analyzing SNPs, they have a blog post here about cleaning up 23andme data and imputing nearby alleles.
When I see an article about some gene that does something, I like to look it up in my own 23andme SNP data. Quite often I don't find the SNP referenced in the article because 23andme didn't cover that SNP, so I'd definitely like to have my whole genome to work with.
That being said, I agree with Dan and Eric. Many of the SNPs I do know about don't predict my actual phenotype (e.g. I have several SNPs that are predicted to make me albino or partially albino ... but I'm not!). Given the low accuracy of the information about SNPs, it is really hard to have any kind of actionable insight from genomics data.
We are really just at the beginning of the genomics age, and we're really waiting on more research correlating detailed phenotypes (e.g. metabolomic profiling) with genomic data. They say genes don't control your destiny -- we just don't know. Does epigenetics control destiny? Transcriptome interactomics?
We thought the human genome would be our blueprint, but at this point it feels more like the genome is a raw binary file, and we don't have the informatics to convert that into the jpeg for the blueprint.