일하다(il-ha-da, ilhada)
To do work or tasks; a general term for 'to work'
When to use
Everyday tasks, Housework, Part-time jobs, Freelance work
The nuance
일하다 can be used in everyday and various contexts, referring to any kind of work including jobs, chores, or part-time work.
근무하다(geun-mu-ha-da, geunmuhada)
To work officially during set hours at a workplace or institution
When to use
Company work, Government office work, Regular working hours, Official duties
The nuance
근무하다 refers mainly to working official hours at formal workplaces like companies or government offices and sounds somewhat formal or stiff in casual conversation.