Difference between cache memory and register

Always - More the better.

If you are talking about the CPU cache, Wikipedia says it best -

A CPU cache is a cache used by the central processing unit of a computer to reduce the average time to access memory. The cache is a smaller, faster memory which stores copies of the data from the most frequently used main memory locations. As long as most memory accesses are cached memory locations, the average latency of memory accesses will be closer to the cache latency than to the latency of main memory.

The L3 cache is simply another layer of cache.

Typically there are now 3 layers of cache on modern CPU cores:

• L1 cache is very small and very tightly bound to the actual processing units of the CPU, it can typically fulfil data requests within 3 CPU clock ticks. L1 cache tends to be around 4-32KB depending on CPU architecture and is split between instruction and data caches.

• L2 cache is generally larger but a bit slower and is generally tied to a CPU core. Recent processors tend to have 512KB of cache per core and this cache has no distinction between instruction and data caches, it is a unified cache. I believe the response time for in-cache data is typically under 20 CPU "ticks"

• L3 cache tends to be shared by all the cores present on the CPU and is much larger and slower again, but it is still a lot faster than going to main memory. L3 cache tends to be of the order of 4-8MB these days.