As Darshana Dewashi has pointed out, বানুর is a corruption (অপভ্রংশ).
To clarify however বানুর is not a corruption of the word বাঁদর, rather they are are both corruptions of the Sanskrit vānar(বানর/वानर), as is the word বান্দর mentioned by Bhuiyan A. Hojaief. The word বান্দর (बान्दर) is the word used in Hindi—perhaps it has come into Bengali through Hindi.
As tomcat pointed out বানর or বান্দর are rarely used—this is certainly my experience living in West Bengal for over 10 years. However, I have to admit I have not spent enough time in Bangladesh for the topic of monkeys to come up in conversation that much ;-)
So it’s safe to say that বাঁদর, বানর/বানুর, বান্দর are interchangeable, but বাঁদর is probably your go to. However as a foreign speaker of Bengali, in the beginning I would find it easier to pronounce বানর—just to avoid the nasal sound.