I think that "Intel Core 2" is the technology and standard that provides multiple cores in a single processor package.
"Intel Core 2 Duo" is then a dual core processor using that technology. You can also get Quad (4) cores; e.g. "Intel Core 2 Quad".
As for "Intel Dual-Core", that's based on an older technology that introduced dual core in a processor (e.g. the "Intel Dual-Core Desktop Processor E2140")
Which is better..."Intel Core 2 Duo" vs "Intel Dual-Core"...that all depends on your needs of price vs performance. If looking at two different computers that have one vs the other, then consider cost, and look online for benchmarks for both processors found within the two PCs you are looking at.
I *think* the above is accurate.
TJR