Generally, a class is Thread Safe when its state remains valid even when its methods are called simultaneously from different threads. Class has to be specifically designed to be thread safe, usually by employing some synchronization mechanism internally, or by making it immutable. Immutable classes do not change their state at all and cannot therefore be brought into invalid state after construction, regardless of how many threads access them. This is one of the reasons why immutable classes are wonderful.
Swing methods are intended to be invoked only from the one and only EDT thread, they have no locking as this was considered slow among other things and multithreaded GUI's hard to design correctly. If you call a swing method from off the EDT thread all bets are off, though it will probably (appear to) work unless they have a guard (don't do it though).
If you say a class is thread safe you can call it from any thread without worrying about synchronization and the Java memory model (gross horrible simplification), in Swing you would use the SwingUtilities to execute your task on the EDT rather than call any method from your thread which your statements claims is ok for AWT (I'd have to check).
AWT is older than swing hence the difference.
"Eagles may soar but weasels don't get sucked into jet engines" SCJP 1.6, SCWCD 1.4, SCJD 1.5,SCBCD 5
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