Is the SPI Interface just a protocol or actual hardware?
SPI is a protocol [de facto] standard developed by Motorola.  It doesn't define any special hardware, only how signals are used and interpreted.

SPI can be implemented using software (known as "Bit Banging"), or using dedicated hardware, sometimes as part of a "USART" interface (Universal Synchronous/Asynchronous Receive/Transmit).

A hardware implementation is invariably more efficient, since the CPU is free to do other things while a byte of SPI data is being transferred.  Also, in chips with DMA the whole SPI subsystem can be completely decoupled from the CPU to run with absolute minimal overhead.

By the way, "SPI Interface" is an example of RAS syndrome.  SPI stands for "Serial Peripheral Interface", so "SPI Interface" would equate to "Serial Peripheral Interface Interface". But I digress.