Vectors will not always be a lighter file size it depends mainly on the information being carried in the svg. For example, I can make a vector shape with a 1000 anchor points in a 200px x 200px file that will most certainly be a larger file size than an equivalently sized png. Similarly, I could shrink an entire magazine illustration to fit into the same 200px x 200px dimensions without losing any information.
I only use these as examples because it helps me answer your question. In all likelihood you are using the right options when saving; however, the complexity of the vectors in the file could be causing the difference in file size.
Since you are using svgs on the web, it might not make sense to have extremely detailed vectors because most users won't ever zoom into the point at which that type of detail matters. There is a simple operation with an object selected in Illustrator: Object → Path → Simplify that you can use as a poor man's technique of removing anchor points in your shape. Hopefully this will allow your object to carry less information without noticeable differences in quality. If that doesn't work you might have to manually rebuild your vector file or accept the increase in file size.