New Copy/Move feature is available in S3 Browser since version 1.9.0.

To Copy/Move files from one Amazon S3 Bucket to another:

1. Select files and/or folders you want to Copy or Move
   Browse Amazon S3 bucket and select files and folders to copy/move.
2. 
   Click Files->Copy if you want to Copy these files.
   Click Files->Cut if you want to Move these files.
   
   Click 'Files->Copy' or 'Files->Cut' to copy or move. You can also find these menu items in the file's context menu or use keyboard shortcuts.
3. 
   Select a destination bucket (and folder if necessary).
   Select a destination Amazon S3 Bucket and folder.
4. 
   Click Files->Paste.
   Click Files->Paste to copy or move file. You can also use Ctrl+V keyboard shortcut.
5. 
   S3 Browser will copy or move the files that you selected to the destination bucket and folder. The files list will be refreshed when all files are copied or moved. You can open the EventLog tab to see detailed messages during copying.
   Files are copied from bucket 'www.images' to bucket 'my-new-images' and 'new-folder-for-images' folder

Alternatively, you can use drag-n-drop to copy/move files between Amazon S3 Accounts and Buckets