"Last night" should be able to replace "on the last night" in most contexts. In terms of meaning, the two expressions are similar. Using "on the last night" emphasizes "the last night" as one of a series of nights: We were there for a week. We camped for the first five nights. On the last night we went to a hotel.
In an example like this, you could also use "the last night" without the preposition: ...The last night we went to a hotel. Using "on" here is just an option.
Here are two examples of this use from TIME*: Radio came up for discussion at dinner the last night of the conference. The last night after the ransom was paid they killed a turkey and gave me a party. It is common to find adverbial expressions of time being used both with and without prepositions although the use of some preposition is usual with the phrase "the last night".
*Time Magazine Corpus