"Phone storage" is essentially your
/data partition, the main space on your phone allocated to apps.
"Internal storage" sometimes means the same but, since they are being contrasted here, it likely refers to the internal "SD card" partition. Likely both of them are on the same flash chip, just set up differently.
SD cards (and "cards") are sometimes mounted later in the boot process, so sometimes it can confuse some launchers and you end up with app shortcuts being removed from your home screen. Services also might not start properly. Otherwise, however, they are more or less the same. You really only need to move apps around if you're running out of space or if you want to be able to wipe one partition without losing as much data.