In Russian one says "в прошлом году", which is in the prepositional, while one says "в прошлый раз", which seems to be in the accusative. Why the difference?
Guess, there's no logical explanation. That's how the language has evolved. Some temporal adverbials with preposition В inflect in prepositional, others do in accusative
в прошлый раз / Новый год / день рождения / месяц / (all weekdays)
в прошлом году / веке / столетии / тысячелетии.
It may have to do with the perceived length of the time span in question. Since prepositional case serves as locative as well, its employment with longer time spans may be thought of as more appropriate because over long periods of time the connection of an object with the time frame could be seen as stronger and so it's kind of placed within one.
The differnce is in context.
You can say both variants "в прошлом году/месяце/периоде" and "в прошлый год/месяц/период", depending on what you mean. But you can't say "в прошлом разу", but only because "раз" is not a period (or place). Think of it as past continuous and past perfect respectevly:
"Этот дом строили в прошлом году"
"Этот дом построили в прошлый год"
You can easily say:
"В прошлый год выдался хороший урожай...",
"Я здесь (without a preposition) прошлый год таких окуней ловил!" (this adaptation I like most of all),
"В прошлом разе оно ведь как было...".
But take account of it's being old-fashioned and rustic but absolutely acceptable in oral speech. These constructions will surely flavour your way of speaking and impart an air of jocosity to it. What had ever been widely used and then became obsolete shouldn't be regarded ungrammatical or irrelevant.