The schools in both communities are excellent. I know much more about the San Carlos School District than I do Burlingame because I have three kids in the system in San Carlos. In a nutshell, the San Carlos School District is outstanding. In San Carlos, elementary runs K-4, and middle school runs 5-8. This is a bit different than other communities, but it works well.
Recently, both of middle schools in San Carlos (Central Middle School and Tierra Linda Middle School) were named California Distinguished Schools. To put this in perspective, there were only 7 middle AND high schools in ALL of San Mateo County to receive this distinction -- San Carlos had 2 of those 7. Here's an article with more details on this subject: http://whiteoaksblog.com/2009/04/01/breaking-san-carlos-scho
If you're interested in the most recent API scores this article outlines the API scores for all of San Mateo County for 2008 -- the 2009 Scores haven't been released yet. http://whiteoaksblog.com/2008/09/05/san-mateo-county-2008-ap
I can't tell you which school district is better. I have friends with children in the Burlingame program and they love it too. But we're very happy with the public school system in San Carlos. No matter which elementary school or middle school you go to in San Carlos, you can't go wrong.
I hope this helps!