I own both, and will attempt to tackle each of your questions in turn:
What are the key differences?
Forbidden Desert removes the 'shrinking map' clock, and adds two new clocks: each player has a canteen with a finite amount of water, and there are limited ways to refill peoples' canteens. Each "Sun Beats Down" card causes everyone (there are exceptions) to have to drink some of their water.
The other new clock is the 'buried alive' clock- where tiles would be getting flooded in Forbidden Island, they instead accumulate more and more layers of sand. If you run out of sand, the game is over.
It also retains the "maxxed out the danger meter' clock.
Replacing the 'shrinking map' mechanic is a 'moving map' mechanic- the game board is basically a 5x5 sliding-square puzzle, and between turns, the map gets shuffled around.
The method for obtaining the MacGuffins is also changed, and player items are one-use-only and not shuffled back into the game.
Is it different enough to be worth it?
In my opinion, yes; absolutely. It's definitely harder to 'win', but is different enough from Forbidden Island to not feel like a reskinned version of the same game.
Which to get first?
If you already are familiar with cooperation-driven games such as Pandemic, I would probably suggest Desert. If you're also looking for a more introductory experience, I'd go with Island. Both games would be good buys in the absence of the other, though, and if you prefer one theme to the other, let that make the decision for you.