Serial versus SPI

The word "serial" doesn't mean much. But I'll assume that you are talking about traditional serial communication standards. What's fundamentally different about SPI is that it is synchronous. As opposed to, say, RS-232, an asynchronous signaling standard.

An important property of asynchronous signaling is the baudrate, the frequency at which the bits in a byte are sent. The receiver has to do extra work to recover the clock that was used by the transmitter. A typical UART does so by over-sampling the signal at a rate 16 times the baudrate. The start-bit is important, which synchronizes the over-sampling clock. Delays between bytes can be arbitrary, the receiver re-synchronizes for each individual byte. Problems with this scheme are a mismatch between the transmitter and the receiver clock frequencies and clock jitter, effectively limiting the baudrate.

This is not a problem with SPI, it has an extra signal line that carries the clock signal so that both the transmitter and receiver uses the exact same clock. And is therefore immune from mismatches and jitter, allowing higher transfer rates. No stability requirements at all in the clock frequency, the signals can simply be generated in software. Typical four line wiring looks like this:

SCLK is the clock signal. MOSI and MISO carry the data, SS is a chip select signal. Common ground is assumed. More about it in this Wikipedia article. electronics.stackexchange.com is a good site to ask more questions about it.

The previous answer is somewhat misleading.

SPI and UART both transfer binary data as bytes and/or words, depending on the hardware. As explained above, one is synchronous and one is asynchronous. Both require an extra data line to be bidirectional. ASCII is an agreed upon interpretation of the binary data and is not actually a factor in either.