All of these interfaces have pros/cons.
UART connection in it's basic functionality requires 2 pins: RX and TX. The SW implementation of how to message over that UART is quite a bit more complicated...you'll have to develop your own messenging protocol between the devices and decide what is a good message and what is a bad message. It could get quite complicated because you pretty much have to define how to "communicate" over the physical link, what is an error, retries, etc. Unless you are implementing a serial port connection to a PC or some other external device, I think a UART is highly overkill for a IC to IC communication path. Master and slave are not specifically defined.
SPI is a master-slave relationship and can be a faster interface (I've seen up to 60MHz clock rates, not common) but it also requires more pins, 3 at a minimum for a point-to-point communication scheme but the number of pins increases to 3+n as the number of "slaves" increases above 1. There are no error indications via SPI. SPI is a "de-facto" standard...meaning it can vary in implementation...your mileage may vary depending on how a IC supplier defined "their" SPI implementation. I generally consider the lack of a true standard for SPI to be a "con".
I2C is also a two pin interface and is an actual "standard" developed by Phillips (now NXP.) As a standard, it is well-defined in how it operates, how errors are raised, and is simple to implement. It has an addressing scheme, can send commands, and can support 0 or more data frames in a transaction. CRC (optional) and higher data rates can be supported (up to 5Mbits.) It does have cons, namely bus capacitance can limit actual data rates (rise/fall time) but generally you can design around this "problem".
In their most basic forms, all of these busses are "ground referenced"...and can suffer from system induced noise. Obviously, lower rail voltages can make this even more of issue. Again careful design practice can mitigate many of the problems some people report to be the bain of their existence.
For the point-to-point system initially asked by the poster, if a master-slave arrangement is required, a SPI or I2C interface may be appropriate (data rate dependent.) If a master-master relationship is required, I2C or UART may be required.
For ease of implementation from a software point of view, I'd rank these communication methods in the following order:
- I2C, if you need faster data rates than I2C can handle, then SPI
- SPI, if you need multi-master, then I2C or UART
- UART as a last resort...has a lot more software overhead to manage the communications channel