Sometimes the argument is from performance. I can buy that.
But the most frequent argument I see is basically, "maybe it'll still work." Basically, it's rude to crash the app on purpose and lose the user's work when maybe, maybe, if you keep on going from an assertion failure the program will still work somehow.
Sure. And maybe you'll crash strangely. And maybe you'll corrupt the user's data, ruining all of their work, not just whatever you didn't save.
What really drives me nuts is that, on the Mac, Apple codifies this crazy idea. The NSAssert macro throws an exception on failure, and uncaught exceptions on the main thread get caught by a handler in the event loop which logs the exception and continues running the app!