You mentioned family of processor. In this case, there is not such thing as "Intel dual core". I believe you meant "Intel Pentium Dual Core"
The difference between the families is the amount of cache, which has an impact in some applications. Cache is memory on the CPU die that is faster than RAM. Obviously, the more the better. In reality, it does not always make a large difference.
Another minor difference is the FSB speed. The Pentium dual core has a slower FSB speed. Not usually a significant difference in most applications.
Here are two CPUs. One is a faster pentium dual core and the other is a slower core 2 duo. The speed makes more of a difference in some tests, whilst the cache makes more of a difference in others: http://www.anandtech.com/bench/Product/94?vs=62
The Pentium Dual Core wins all but 5 of the benchmark tests. You can also see that one type of application that really uses the cache is gaming.
It's important to remember what applications you intend to use when purchasing a new CPU.