Budo : martial arts. the code on which martial arts are all based.
Origin: Japanese budō, from bu ‘military’ + dō ‘way’
Butoh : a style of Japanese modern dance
Origin: Japanese literally “to dance”
From Oxford Dictionaries
These two words sound similar to me, and I think that they have similar foundations. Dance (to my mind) expresses a natural energy flow within the dancer. The code which martial arts sits upon (to my mind) expresses itself as a natural energy flow within the martial artist. The similar soundingness seems to say so much more, something like “pay attention to your own natural energy flows and express them”.