Compatibility between Arduino and Arduino clones
Over the years I have used original Arduino boards as well as a multitude of clones of varying provenance, from the excellent InduinoX that is listed on the Arduino site as an official compatible board, to the very well-made Chinese Meduino Nano Enhancement (3.3 / 5 Volt switchable), to dirt-cheap clones from eBay that are sometimes better constructed than the official boards.

There is nothing unique or sophisticated about the Arduino board: It's basically just the microcontroller (atMEGA328, atMEGA1280, SAM3X8E or others) and its minimal support hardware. Not much that can go wrong with that little, really. 

The two common elements that ensure compatibility across these boards are:


The Arduino bootloader
Support for the board within the official Arduino software distribution


The bootloader can be replaced with a better or more up-to-date version if one is available for the microcontroller on your board.

The support files can either be found online, or minor edits to the nearest similar board's files can provide the requisite support for the Arduino environment. 

In short: Yes, clones typically work just as well as the original Arduino, and clones with enhanced features often work far better while retaining compatibility (e.g. InduinoX)