What are the differences between gumbo, jambalaya, and etoufee

It's actually spelled 'roux', and is a mixture of oil and flour, cooked to remove the starchy taste of the flour.

It's a great thickener any time that you don't need the sauce to be clear, and you have time to cook it down. I typically use it for cream sauce (including cheese sauces, such as for mac & cheese) and gravies.

As for benefits -- it's habit at this point, so I'm not really sure -- I guess it's made from things I always have on hand. You can also get flavor from the roux, if you cook it longer , but you'll adversely affect the thickening ability. Gumbo is normally made from a dark roux (the Cajuns have a series of names for the color of roux, including 'brick', 'peanut-butter' and 'chocolate')

The previously accepted answer contains a good link, but not any information here on Seasoned Advice, so I thought I would write out a few things from my understanding after having developed an abiding interest in Cajun foods over the past decade.

Basically, Creole is a much broader term with a longer history. While it is perhaps inaccurate to say Cajun is a 'subset' area and culinary style, it is more geographically- and flavor-specific than just saying Creole. The history of people and region identified as Creole (on the American continents) goes back a long ways, and Cajun is a more recent phenomenon. Creole can also refer to other peoples and regions other than just in the South/Central parts of North America, though such use in the United States is a distant memory.

In the past century, the "Cajun" term has certainly attained an independent status as a self-identified people, region, and flavor -- the last no doubt owing to the flavor potency of their offerings. Which offerings, in my opinion, when consumed, are nearly always exquisitely enjoyable examples of perfection in the experience of food.