I am taking a course in stochastic process this time. As I read through a couple of books every one mentioned that stochastic process is also a random process. So, my confusion is why we call stochastic process just for a random process? For me random process makes a lot more sense than the stochastic process. I asked this question to my professor too but he replied me "I don't know."
This difference fall into context and perspective. In mathematics, in general, they are synonyms, but it can be thought as different concepts from an epistemological point of view, depending on the case.
A possible example of the later: stochastic may refer to a not well known process (maybe deterministic on some degree) that is modeled as a random process.
An example could be the climate: it can be modeled as a random process or as a deterministic chaotic process, or a mix of both.
Stochastic process and random process are just two different names for the same mathematical concept, which are used interchangeably, just like convergence in law and weak convergence, and there is no objective argument for or against one or the other. In analogy with the english language, you can interpret it as dialects (the British say rubbish, Americans trash). The most important thing is that you understand the concept.