There were two main reasons for this. First, the AF of L was less radical than many of the other unions which did fail. Second, the AF of L generally represented more skilled workers than the other unions did.
One reason why many unions failed was public opinion. These unions, like the Knights of Labor or the IWW, were often opposed to the entire capitalist system. The unions were also connected in the public mind to violent incidents such as the Haymarket bombing. For these reasons, the public was not sympathetic to these unions and did not support them.
Another reason why unions failed was more directly economic. Unions have more power when their workers are less replaceable. The unions that failed were often unions that tried to represent even unskilled laborers. In the late 1800s, unskilled laborers were very common in the United States. A firm that fired unionized workers who were striking could easily find replacements. By contrast, the AF of L represented more skilled workers who were harder to replace. This meant that the AF of L was more likely to be able to force employers to negotiate.
For these two reasons, the AF of L was able to survive even when other unions failed.