What difference between I7 processor and Intel Quad-Core???

simple way to look at the difference is a quad core processor is designed to run 4 different processes at the same time, one on each core, the i7 processor can do this as well as usually there are four cores as well but the I7 also has more technology built into it such as Hyper threading of which a quad core does not have, so each core in a I7 has the ability of running 2 threads within each core so you then have the ability to run two task on each core, four cores, two tasks on each you then can efficiently run 8 different task in total.

you need to decide what you want to run to decide on your needs, the I7 can run more programs at the same time then what a quad core can.