The simple difference is that conv() operates on a 1D vector, while conv2() operates on a 2D matrix.
conv() is usually used in the study of linear system analysis and control theory. The most common application of conv() is to find the response of an LTI system. If the impulse response of an LTI system is given, then finding its response to any input is just a matter of finding the convolution of input and impulse response. See these links for more information.
conv2() is mostly used in image processing. It can do a lot of things with images, e.g., blurring, smoothing, sharpening, etc. It is impossible to summarize its application in a single post. Following links will be helpful