Those two devices are very different. Depending on exactly what you're trying to do, you may be able to use either, however. You'll have to tell more about your goal.
If you're switching high-current or high-voltage loads on and off, you'll need some sort of relay (or perhaps a large FET). If your current and voltage requirements are sufficiently low (5V, 40ma), you may be able to drive your load directly with the Arduino's output pins.
The Arduino is a microcontroller. That means it's an entire computer, just simplified. It has RAM, registers, an ALU, etc. Microcontrollers are generally specialized such that instead of interfacing to peripherals using some kind of bus like in a desktop computer processor, they have I/O capabilities built in, often simply in the form of outputs that can be set high (the input voltage, usually 5V) or low (0V) programmatically. The Arduino probably uses its own programming langauge, although there may be more than one language available for it (I've never used one). I doubt PHP is one of those langauges.
The relay controller is exactly what the name implies -- a simple circuit that controls some relays. Relays are electrically actuated switches. There's no intelligence in the relay controller. It can't be programmed; it must be controlled externally via USB. If you're attempting to interface with it from PHP on a desktop/server computer, this is probably your best choice. You're right that it's expensive. You could probably build your own for a fraction of the cost, especially if you're willing to use the parallel port on your computer (googling for how should give simple instructions). It's worth noting that that relay controller, and presumably most others, likely contain some kind of microcontroller with the I/O pins connected to circuitry that increases the current and/or voltage to the point where it can drive the relay, which in turn switches the load.
Hmm... only very vaguely programming related :) I think we may need another StackOverflow for electronics. Maybe SparkOverflow?