Could any one tell me what is exactly done in both situations? What is the main cost each of them?
The main distinction between a thread switch and a process switch is that during a thread switch, the virtual memory space remains the same, while it does not during a process switch. Both types involve handing control over to the operating system kernel to perform the context switch. The process of switching in and out of the OS kernel along with the cost of switching out the registers is the largest fixed cost of performing a context switch.
A more fuzzy cost is that a context switch messes with the processors cacheing mechanisms. Basically, when you context switch, all of the memory addresses that the processor "remembers" in its cache effectively become useless. The one big distinction here is that when you change virtual memory spaces, the processor's Translation Lookaside Buffer (TLB) or equivalent gets flushed making memory accesses much more expensive for a while. This does not happen during a thread switch.