It's obvious dual core has two
and the quad core has four processors:
But yeah, how the fuck are they different, good question btw:
The no. of cores are different but, the cores are embedded on the same hardwar. So, they perform the functions on the data in the same memory and path to the motherboard.
Quad core can execute functions four times faster but, it doesn't make them two times faster than dual cores because they recieve instructions with the same speed as of the motherboard.
So my suggestion is to take a phone that suffices your purpose.
If its not to graphical, go for a dual core.
Also if your usage takes up a lot of graphic functions,
and if you plan on taking a quad core mobile, know this:
this engages a lot of battery for functions, being faster processors,
they also generate a lot of heat that causes battery problems,
So take the one that has atleast 2 gb ram and above 3000 mAh battery.
Maybe give us a little more information about those two phones. I find that quad-cores are generally faster, from my experience with phones, but I wouldn't think this is the case for absolutely all phones. Maybe give us the specs for the phones and then we could hep you more.
I'll try to give a more general explanation. A dual core device is when the device's processor has two cores to use. Generally nowadays, dual cores in the Android world are very much outdated as Android has grown to be a very complex system in which devices which have dual-core chipsets have a hard time to handle Android smoothly, unlike those days.
A quad core device is basically a device which has 4 cores in it's processor. As you can see, a quad-core device has more breathing room to which it can distribute a task not limited to a single core, but to the rest of the cores as well. This depends on the complexity of the task. As all these cores work together to do one task, they can pretty much process the task faster.
So, the conclusion for your question is, you should definitely go for a quad core device.