Both fulminant and acute are adjectives that describe a disease course. The previous answer by @ilan is incorrect. Fulminant does not specifically refer to adrenal failure, despite the association between meningococcemia (which often has a fulminant course) and adrenal hemorrhage.
Fulminant relates to onset, speed of progression, and severity. A fulminant disease process is sudden, rapidly progressing and very severe. Diseases processes described as fulminant are often lethal if not immediately treated. Meningococcal bacteremia, classically described as purpura fulminans, is one example.
Acute relates mostly to onset. Here the earlier answer by @ilan is correct. Acute, as opposed to chronic, refers to a disease process with a short course, or a short term phase of a disease process with a longer course (as in an acute exacerbation of a chronic disease process).
All fulminant disease processes are acute, by definition. When referring to bacteremia with Neisseria meningitidis, the use of the descriptor acute or fulminant does not indicate a different disease. So I suppose the direct answer to this question is that there is no specific difference between fulminant and acute meningococcemia. They are not separate diseases, and both terms could correctly be used to refer to the same case.