Yes, the methods differ, as do the assumptions, theoretical commitments, standards of education, traditions, styles of writing, and so on. But they're not sciences, so tribal differences are allowed to impact the fields. What's your point?

Mine would be that the notion of "method" is watered down when dealing with nonsciences. "Method" gains an exact meaning only when you're dealing with a science. So the "methods" of the different philosophical schools will be tainted with politics, literary fashions, elite prejudices, and so forth, which make the endeavours more subjective and artistic than objective and scientific.