The core count is the physical number of cores on the CPU die itself, whereas the thread count is the number of individual application threads which can be executing simultaneously on the CPU itself. Without any additional or special hardware, this is equal to the core count. Some processers, however, will have more threads than they have cores.
Some Intel CPUs have a feature called hyperthreading, which allows an operating system to see double the amount of logical cores per physical core. This allows the operating system to schedule and run double the amount of threads simultaneously, so in the case of the CPU I linked to above, there are four physical cores, but eight logical ones (so you can run eight threads simultaneously).
Each individual application running in the operating system is either single-threaded or multi-threaded (think of each thread as a "sub-application"). Single-threaded applications require just one thread to run on the CPU, whereas multi-threaded applications have many sub-threads running simultaneously. Additional cores, or hyper-threading, allow more application threads to simultaneously run at once.
This allows multi-threaded (not single-threaded) applications to run much faster, since more than one thread can run at once on the CPU.
Just a final note, hyperthreading improves the performance of some multithreaded applications specifically optimized for it (since there is still only half the number of physical cores as there are logical). In some various cases, applications may run faster with hyperthreading disabled (although many applications do benefit from it). Regardless of hyperthreading, an increase in the number of physical cores will always benefit multithreaded applications.