Yes, both speak the Korean language. However, over the decades since the partition of Korea, there has been a degree of divergence. This article gives a rundown of some of the linguistic mixups North Korean defectors to South Korea encounter.

They do, the current division of north & south korea is something that occurred due to external politics (specifically post-WW2 politics) rather than internal causes. Generally speaking, the korean peninsula's been united into a single kingdom since around the 7th century AD.

Basically the language is the same with a few exceptions:

• Regional differences. Every region in korea has its own differences in accent and terminology. The best analogy that comes to mind are the regional variations between different regions in england. Within south korea the jeju dialect is the most difficult to understand for non-jeju people.

• Loanwords. Since independence in 1945, south korea has had an influx of english loanwords while north korea has tended to create its own replacement words. For example, computer in south korea is simply "com-piu-to" (i.e computer literated into the korean script). This was accelerated as learning english became more prominent for schoolchildren.

Basically there's differences but they're mostly in terminology and someone from north korea should only take a short period of adjustment (say a few months).