IplImage has been in OpenCV since the very beginning. It is a part of the C interface for OpenCV. You need to allocate and deallocate memory for IplImage structures yourself. (remember the cvReleaseImage commands?)
The new Mat structure is a part of the C++ structure. So obviously it is object oriented. Also, it manages all memory for you! It keeps a track of references to it. Then the number of references goes to zero, it deallocates automatically. This is one superb feature!
Go for Mat. It should be easy to translate code from the IplImage thingy to the Mat thingy if you are using some IDE that has Intellisense (it drops down a list of possible functions, variables, etc as you type)