When you connect the phone to your PC, it will be recognized as another external drive on the computer. Then select the picture folders and copy them to one of your hard disks.
Next, disconnect the Bionic from the computer and hook up the S4. Wait for it to be recognized as an external hard dis and then reverse the process.
Pretty standard copy and paste stuff. It will take a few minutes for the phone to be recognized as a drive. Just have a little patience while the PC goes through the process the first time of recognizing the phone.
If you have a SD card, the phone will contain two drives: one for the internal device storage and another for the external SD card.
You could user Bluetooth to transfer them from phone to phone without a PC involved. Bluetooth File Transfer in the play store does this. It can be a bit slow, but does it without a PC. I have also used my Network Attached Storage(NAS) on my home network. Move the pics from the Bionic via wifi to the NAS then move them to the S4 from the NAS via wifi. It is faster than Bluetooth.
All my pictures were on the sd card of my Bionic. Since my S4 didn't come with a card, all I did was move the card over. The S4 saw them without a problem.