Rather than giving the full list of specifications of the devices, I would prefer offering a real kind of comparison based on some important criteria.-
Design and Build Both the Devices in question are closer to each other. With the physical dimensions almost similar, Redmi Note is a bit thicker. Also it weighs 199 grams while Yu weighs only 177 grams. Design and build quality is quite solid on both. Redmi feels a bit more durable because of its weight. Yu is slightly sleeker and lighter. The matte finish on Yu appeals more than the glossy plastic on Note. Yu fits better in your hand while the Note feels heavy.
Display Both the devices sport a 5.5 inch screen with 720p resolution. Colors look little warmer on Redmi Note. If you are fond of rich colors, you will like Redmi. But the screen of Yu is more balanced. It also has better outdoor legibility. Note on the other hand has better viewing angles.
Software Yu runs on Cyanogen mod 11 OS - based on Android Kitkat. Redmi Note is an Android device with Kitkat 4.4 running on it with MiUi skin. MiUi is more easier to use. It differs much from the stock Android experience . It provides you iOS kind of experience. Cyanogen on the other hand is almost stock Android. But it has pretty many customization options. This allows you to truly customize your phone.
Hardware So finally it boils down to the performance to decide the better one among these two. Yu is powered by Snapdragon 615 Octacore processor clocking at 1.5 GHz and Adreno 405 GPU. Redmi is powered by Snapdragon 400 Quad core processor with a clock speed of 1.6Ghz. It has Adreno 305 GPU. Internal memory provided on Yu is a whopping 16 GB while Redmi Note offers just 8 GB. Both the devices offer almost similar lag free performance. When it comes to sound quality, Note has better speakers - however the difference is too subtle. With its Octacore processor, Yu looks future proof. Both devices can handle resource hungry apps quite easily - thanks to the 2 GB on board RAM on both the phones. The memory can be expanded up to 32 GB on Yu while Redmi allows memory expansion up to 64 GB.
Camera The similarities continue even here. Both of them have 13 MP rear camera with Auto focus and flash. The front snappers are 5 MP. Redmi can capture better colors. But the images captured on Yu have more details. The metering is better handled in Yu.
Battery Yu has a 2500 mAh battery while Redmi uses a 3100 mAh one. It is found that Redmi gives more standby and talk time - almost two and a half hours better than Yu.
My Choice As I have compared the devices above based on some key points, it is found that either of them almost similar. However I would prefer Yu because of its better and newer processor. I treat it slightly better due to the more internal memory and support for dual SIM. Also it is cheaper by around Rs. 1000 than the Note.
Please note that I have not indulged in pointing down each and every feature of these devices because they have been well explained in the previous responses to similar questions. I have only pointed out the most relevant points for comparison.
Live....and Let Live!