Learn French: En dépit de notre aide, la police a démasqué l'informateur. - In spite of our help, the police unmasked the informant.

They mean exactly the same thing ("in spite of" or "despite") and both must be followed by a noun or pronoun—in this case, the noun phrase notre aide. However, en dépit de is considered slightly more formal and literary than malgré.

Could I say en dépit que if I wanted to follow it with a conjugated verb instead of a noun?

No, en dépit que does not exist in standard French! En dépit de must always be followed by a noun or a pronoun. If you want to express the same idea using a conjugated verb, you need a conjunction like bien que or quoique (meaning "even though" or "although"), which must be followed by the subjunctive mood.

Why is the verb singular (a démasqué) when "the police" usually takes a plural verb in English?

In French, la police is a strictly singular collective noun. It refers to the police force as a single institution, so it always takes a third-person singular verb. If you want to talk about multiple individual officers, you would use the plural les policiers.

Would de change if it were followed by a plural noun phrase instead of notre aide?