My understanding is that synthetic biology is genetic engineering 2.0. The difference is in the approach. Whereas genetic engineering projects are usually ad hoc, synthetic biology aims to apply proper engineering principles such as standardisation, modularisation, and reusability. Synthetic biologists create and use libraries of standard parts that are characterised, so they can be easily reused in projects. A part could be a gene, a terminator, a promoter, etc.
Synthetic biology also has greater ambitions. The focus is on creating whole systems/circuits of genetic regulation. This means there is a need for computational modelling and understanding of how biological systems work. In this aspect synthetic biology is a sister of systems biology a bit like synthetic chemistry (engineering) is a sister of chemistry (science).
You could of course argue that it's just a marketing ploy to invent a new name for something that is just the next step in genetic engineering, but the differences in approach are quite large and a new name signifies it.
With regards to synthesised vs. PCRed DNA: It doesn't really matter which you use in synthetic biology. However, cheap synthesis is one of the technologies that enable easier synthetic biology. The idea for the future is that you will be able to synthesise whole plasmids and chromosomes instead of having to "cut and paste" DNA. When that happens physical parts repositories will be obsolete, but they will remain crucial in silico. Cheap synthesis is nice, but doesn't make or brake synthetic biology.