Difference in Interpretation: $ω$ (rads/s) vs. $ω$ (rads) and $X(ω)$ vs. $X(e^{jω})$
Most people learn (or try) DSP from this book


  Oppenheim, and Schaefer, (1999). Discrete-time signal processing. Pearson Education.


and there $\Omega$ is used for continuous time and $\omega$ is for discrete time, which is how they keep their notation consistent. Usually both are continuous, which can lead to some confusion when you consider the DFT and/or Fourier series.

In the Olden Days, Yea DSP was primarily practiced by people who had a background in continuous time signals derived from some physical problem, and notational conventions for $\omega$ were typically $2\pi f$ (in yonder rads/sec).

Most books back then, tended to start with some general Linear Systems theory, and then go to sampling.  I think the most recent version of Oppenheim and Schaefer wait until Chapter 4 to cover sampling.

People who taught and wrote about DSP, recognized that including a sample rate wasn't really necessary.  My first DSP class was taught by a guy who did adaptive beamforming and he introduced the concept of unity sampling, $f_s=1 \text{ sample/second}$ so all our filter frequency responses were plotted between -1/2 to 1/2 Hz.  You just needed to scale everything by your actual sample rate when you got back to your physical problem.

It became clearer over time that there were time series, like Stock Market time series that were inherently discrete in time and a lot of DSP didn't really require reference to time, and things like image processing really didn't either, so to make things more general, $\omega$ as radians seems to have become the most common notation. You can also sell your books to finance majors now.

The notation $X(e^{j\omega})$ is handy when you plug $e^{j\omega}$ into a Z transform (if the unit circle is in the region of convergence), and reinforces the concept that a function or composition of a periodic function $e^{j\omega}$, is periodic.

The notation $X(\omega)$ needs to be interpreted in the context of where it is, could be discrete time, could be continuous time. 

So, the answer to your question is: Context.  

As an American English speaker, it would be natural for me to assume that this is the complete story but someone who learned from the Russian (or French, or ....) DSP literature would most likely have a different explanation.