Dual-core processors are processors that contain double cores in one silicon chip, while core two duo is the name that Intel gave to their second batch of the dual-core. The manufacturers of processors like AMD and Intel have gone a different route to improving the performances of their products, by introducing two cores into a single chip so as to boost the performance of the OS. The two cores present in the chip will be able to multitask without hindering themselves, this leading to delivering better and faster performance.
Initially, AMD usually referred to their dual-core processor as X2, while Intel referred to theirs as core duo. Intel advanced into refining their product by having 2 Pentium 3 processors in a single chip, and they decided to make the name core two duos, and not duo core anymore. Every other dual-core processor apart from the core two duo are the X2 series from AMD or core duo from Intel.