I took your code and created the following sketch (I did not have a data folder)

1. String[] fileNames;
2. 
   
3. void setup() {
4.   size(600,500);
5.   background(255);
6.   fill(0);
7.   java.io.FilenameFilter txtFilter = new java.io.FilenameFilter() {
8.     public boolean accept(File dir, String name) {
9.       return name.toLowerCase().endsWith(".mp3");
10.     }
11.   };
12. 
    
13.   fileNames = listFileNames(sketchPath, txtFilter);
14.   text("MP3 files in " + sketchPath, 20, 20);
15.   if(fileNames != null) {
16.     for(int i = 0; i < fileNames.length; i++) {
17.       text(fileNames[i],20,60 + i * 16);
18.       println(fileNames[i]);
19.     }
20.   }
21. }
22. 
    
23. String[] listFileNames(String dir, java.io.FilenameFilter extension) {
24.   File file = new File(dir);
25.   if (file.isDirectory()) {
26.     String names[] = file.list(extension);
27.     return names;
28.   }
29.   else {
30.     return null;
31.   }
32. }

I then copied some mp3 files into the sketch folder and ran the sketch from Processing and it displayed the list of filenames.

I then exported as an application (Windows) and copied the mp3 files into the applictaion folder where the .exe was created and again it worked.

I then exported as an applet and copied the mp3 files into the applet folder with the index.html which I then opened in Firefox and all I got was a blank screen, not even the output from line 14 displayed anything.

It could be something to do with security since applets are more restricted than applications but it has me stumped.