The answer largely depends on the specific country/party; socialism is a scientific ideology, after all, and often experiments with new ideas or processes depending on the specific conditions of that country and its needs.

Generalizing though, the party is an ideological animal whereas the government/state is a practical one. The latter concentrates on day-to-day issues like infrastructure, education, the economy, etc while the former acts to guide the state towards the goals of socialism. As a practical example, the government may be working to expand light industry to create more luxury goods for its people while the party would be working to ensure the long-term benefits of such go to the working class and not get consolidated into the hands of a wealthy minority. Both the party and the state are tightly integrated to ensure that this isn't just a bunch of armchair Marxists reading theory and yelling at a government that largely ignores them, so you'll often find that party membership is essentially required to get into the state in the first place (though there are, contrary to popular belief, multiple parties within typical ML governments. China, North Korea, Cuba, etc all have multiple parties, just with a very dominant communist party, so there is some wiggle room here).

The confusion around long-lived leaders generally boils down to this separation of party and state: a populist figure like the Kims might start off as both head of state and head of party, but gradually shift duties more towards the latter until they completely abandon the head of state position. Since the party still has massive influence this means they still have quite a lot of sway, but they're not making the day-to-day decisions directly anymore.