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I did a search on Kayak for an upcoming flight and it's price trend suggests waiting, citing 60% confidence that the price of the ticket will do down. However when I search for the exact same flight on Bing Travel it suggests buying the ticket, citing 80% confidence that the price of the ticket will go up. The real stumper is that Bing's results are powered by Kayak.

Does each site use a different algorithim to determine the price trend? Does anyone have any advice on which is more accurate?

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Yes, they use different algorithms.

Bing made the switch to use Kayak back in 2011, and their partnership meant that:

Bing Travel flights has retained its “look and feel,” but with the exception of Bing’s Price Predictor, the tools and filters basically are the same.

Recently, Kayak produced its own new price forecaster, and if you'd like to read more about how that works, they have an article on their site: KAYAK Launches Price Forecasts For Flights.

But yes, long story short - different forecasting algorithms, and pretty much the same flights.

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