No, assembling won't tell you anything new. When working with organisms whose reference genome is known, all you care about is where your sequences differ from the reference. If you know where your sequences differ, then you can infer the rest, they will be whatever is in the reference.
All you want to do is identify variants, those sites where your genome differs from the reference. That's what the VCF file contains. Bear in mind that the reference isn't some sort of platonic ideal of the perfect human, it's just the collected, pooled DNA of 13 anonymous, apparently healthy individuals. You can expect anywhere from about 3 to 5 million differences in your own genome, the overwhelming majority of which will be completely innocuous.
Since you already have your VCF file, your list of variants, that's all you need to investigate. If you really want to, you can make your assembly by taking the reference genome and then adding the variants in your VCF file, but it won't be of much practical use.