Everybody agree to tell that... new String("literal string") is quite useless!
Not only that, but it has unwanted side effects!
Note this code:

1. String S1 = "Processing";
2. String S2 = new String("Processing");
3. String S3 = new String("Processing");
4. String S4 = String.valueOf("Processing");
5. void setup()
6. {
7.   if (S1 == "Processing")
8.     println("Works with S1");
9.   if (S2 == "Processing")
10.     println("Works with S2");
11.   if (S4 == "Processing")
12.     println("Works with S4");
13.   if (S1 == S2)
14.     println("Works with S1, S2");
15.   if (S1 == S3)
16.     println("Works with S1, S3");
17.   if (S2 == S3)
18.     println("Works with S2, S3");
19.   if (S1 == S4)
20.     println("Works with S1, S4");
21.   if (S2 == S4)
22.     println("Works with S2, S4");
23. }

You will see that S1 and S4 are the same object, while S2 and S3 are different objects (the == compare references of objects).
That's because Java tries and optimize string usage: when it finds identical literal strings, it just create them once and assign the references to the variables. It works because strings are immutable, so if you "change" S1, S4 won't be changed.
String.valueOf() do the same, it is called "string interning".
Note that something like Integer.valueOf() is interesting too (it returns an Integer object) because if you create lot of objects with int value between 0 and 127 (IIRC), it will create only one instance of each, reusing the objects. Good to know if you put ints in an ArrayList, for example.

Note that's the code above shows the only case where you can compare strings with ==, when they are literals in the same class (or same program? not too sure on this one).