A UART (Universal Asynchronous Receiver-Transmitter) is a particular kind of hardware device that uses digital sampling to convert serial data that uses embedded start and stop bits for synchronization into and from parallel data. Not all forms serial communication use start and stop bits, and not all devices that use this form of serial communication uses some sort of UART hardware. A serial port could be something completely different.
In traditional asynchronous serial communication, data is transmitted a series of "mark" and "space" values. How these values are represented depends on the communication medium, for example RS-232C defines a mark as being a voltage in the range of -3V to -15V, while a space is +3V to +15V. In order to know where each byte (or character) starts and ends, each byte is prefixed with a space value (start bit) that indicates the start of the byte, and suffixed with one or more mark values (stop bits) that indicate the end of a byte. In between the start and stop bits each byte is transmitted as series of mark (1) and space (0) values.
The start and stop bits provides the necessary synchronization for this form serial communication to work by just transmitting mark and space values without needing precisely synchronized clocks or by also transmitting a clock separately. The start bit lets the receiver know that a fixed number of bits will follow at given rate. The transmitter and receiver must be configured before hand to use the same parameters, the same number of bits in a byte, the same number of stop bits and the same baud rate. Once the receiver has received the start bit it samples the communication line at the specified baud rate to obtain each bit of the transmitted byte in sequence.
The receiver ignores the the stop bit, but it performs two functions. First it gives time for the receiver to prepare for the next byte, and this was crucial on old mechanical devices like teletypes. Secondly it guarantees that the start bit, a space value, is preceded by a mark value, meaning each transmitted byte is preceded by transition from a mark to a space. This allows the receiver to resynchronize with the serial data byte steam if starts receiving in the middle of a byte being transmitted or noise or other problem causes it to lose synchronization.
Since the start bit in fact causes the receiver to resynchronizes with the transmitter every time a byte is transmitted this means the receiver doesn't need to be sampling each bit at precisely the same rate as the transmitter is sending them. Since normally only 10 or less bits, including start, data, and stop bits, are transmitted before this resynchronization occurs even a relatively large 5% difference in transmitting and receiving baud rates won't cause errors.
A UART does everything necessary to implement asynchronous serial communication in hardware, and does so digitally (other earlier asynchronous hardware implementations used analogue timing circuits that needed manual calibration.) However it's also possible to implement an asynchronous serial device in software using a technique called bit banging. This was done on old computers like the Commodore 64 in order to save money, and is still done from time to time in modern embedded applications for the same reason.
Finally there are other ways of transmitting data serially, like synchronous serial communication methods where a separate clock signal is transmitted alongside the serial data. In fact many modern computer interfaces, like USB, Serial ATA and PCI Express are actually serial devices. So while a "serial port" normally refers to an asynchronous serial device usually implemented with some sort of UART, strictly speaking it could potentially mean a wide range of devices. By calling the hardware device used a "UART", and the software driver for it a "UART driver" the diagram you've shown in your question is being more precise about what its describing.