Along with the bannou knife and santoku knife, the bunka knife (or bunka-bōcho) is an all-purpose kitchen knife for meat, vegetables and fish. 'Bunka' is the Japanese world for culture and when the name of this knife was coined Japanese food habits were evolving quite quickly, along with the many other changes to society, technology and infrastructure after the second world war.
It was around this time that the bunka, bannou and santoku knives all became commonly available. There's nothing to differentiate the three, so if you have one in mind it's best not to ignore the others. Each has a moderate blade thickness appropriate for meat and fish but thin enough for easy processing of vegetables. Not at thin as a nakiri, but thin enough not to split your squash.
Knife makers in Japan use different steels, different cladding, different profiles on knives with similar uses, in order to provide variation in their line-up and maximise sales. So it's not unusual for a knife maker to include, say, a bunka and a bannou in their catalog. The different names are simply a means of differentiating between the all-purpose chef's knives on offer. One bunka will vary from another in style, blade length, height and profile, and may come with a western or Japanese handle.