Portal is noted for having a dark-but-hilarious story line. Quantum Conundrum is a lighter tone, where you must rescue your Uncle from his own inventions. Both feature funny narrators, the key difference being one is a crazy Uncle guiding you to save him, and the other is a robotic artificial intelligence that is running grim science experiments on you.
Quantum Conundrum's main gameplay hook is that you jump between parallel dimensions to solve puzzles. By using the Inter- Dimensional Shift Device, it is possible to switch between 4 different dimensions (or 5, if you count our dimension) to help you solve the puzzles of the mansion you are in. The hook of Portal was to use a Portal Gun to go from Point A to Point B. The portals aren't just limited to the player, allowing for other items to also be sent through. Using the Portal Gun, the player has to make it all the way through the series of test chambers they are stuck in.
One of the most memorable parts of Portal is the overbearing GLaDOS. The parallel to GLaDOS in Quantum Conundrum is Professor Fitz Quadwrangle, your inventor Uncle. Both serve as the primary story-tellers of the games, and both are mostly just narrators without actually being on the screen.