Is there a difference between right-tailed ADF and standard ADF tests in Eviews?
Yes there is a difference. The right tailed test in Eviews is designed based on: Caspi, 2013.
If you're looking for just the operational difference between the two tests: The standard ADF test is also one sided with $H_0: \phi = 0$ against $H_1: \phi<0$ (so in practical terms the null hypothesis is $H_0 \geq 0$). Rejection of null would mean no unit root and the series is stationary.
For the right tailed test the alternative is $H_1: \phi >0$. So the rejection of null means that the process has no unit root but is non-stationary (explosive type).
I do want to mention that I do not know theoretical difference as in how would one go about designing such a test because the null distribution is same but the critical values for different alternatives need to different. But how, I don't know.