Theoretical physics is the field that develops theories about how nature operates. It is fundamentally physics, in that the ultimate goal is to describe reality. It is informed by experiment, and at the same time it extends the results of experiments, making predictions about what has not been physically tested. This is accomplished using the language of mathematics, and often the demands of theoretical physicists force mathematicians to extend this language in new directions, but it is not concerned with developing the language of math. Theoretical physicists are, among other things, physicists who are very well-versed in math (which is not to say other physicists are not - please don't hurt me).
Mathematical physics, on the other hand, is a branch of mathematics. It explores relations between abstract concepts, proves certain results contingent upon certain hypotheses, and establishes an interlinked set of tools that can be used to study anything that happens to match the relations and hypotheses on hand. This branch in particular is motivated by the theories used in physics. It may seek to prove certain truths that were simply assumed by physicists, or carefully delineate the conditions under which certain theories hold, or even provide generally applicable tools to physicists, who can in turn apply them to nature. Mathematical physicists are mathematicians who are intrigued/inspired by physics.
One could say that mathematical physics is concerned with the internal, logical consistency of physical theories, while theoretical physics is concerned with finding the right model to describe the world around us. Very roughly, one might diagram these things as shown below.
$$ \text{Mathematical physics} \Longleftrightarrow \text{Theoretical physics} \Longleftrightarrow \text{Experimental physics} $$
Actually, one wave and its 180 flip phase image can be seen as two waves with the same phase but with opposite amplitude (when one is positive the other is negative).
The reason you cannot hear the difference between the two is that your ear (or microphone) is not sensitive to the amplitude but rather to the intensity of the wave, i.e. the square of the absolute value of the amplitude. As you can see, the difference in amplitude sign (or the phase) does not matter and both the original and its 180 flip phase image will sound the same.
What you would like to measure to see the difference between the two is the phase of the waves but as mentioned earlier, you cannot detect the phase directly. However you can measure the interference between the waves : when waves are emitted simulatenously they sum up. Now if the two waves have the same phase, then the amplitude add up (constructive interference). If the phases are 180 degree apart, then the amplitudes will cancel out (destructive interference, mentioned above). If the phase difference is somewhere inbetween, then you will get something ... inbetween (an interference pattern).
Best Answer
Both terms describe disturbances in some medium. Wave usually refers to a continuous disturbance. Like if you grab hold of spring and shake it back and forth a lot. Pulse, on the other hand, often refers to some type of one-time disturbance. Like shaking the spring only once.
Of course there will be overlap or ambiguities in these terms. I doubt there's any agreed-upon precise definition of these.