The internet controllers are chipsets that perform the entire task of physical handshaking the signal, as well performs the logical negotiation of the protocol, and operate as 'slave' devices, whereas built-in Ethernet controlers are nothing more than these same devices, but now embedded at the IC of the uC. For some reason, it seems that the current trend has been the core manufacturers no longer integrate Ethernet controllers on the same chip.
I particularly liked a lot the Luminnary ARM series LM3S6965 ( which now belongs to TI ) that had either the MAC and PHY control at the same device, requiring no extra external ICs to perform the ETH connection. However, such uCs unfortunatelly are now labeled as "not recommended for new designs".