Deciding if something's realistic is strongly related to science, isn't it? What's the essential difference between these tags? When they don't come with each other?
The tags description pretty much tell you all you need to know.
Unlike science-based and hard-science, reality-check question do not require a scientific explanation. The approach for questions with this tag should be to ensure logical consistency. In short, does the concept make sense given the rules specified?
Questions with this tag should be answered as far as possible based on known scientific fact or reasonable extrapolations from that, but answers are not required to provide scientific citations. For example, we do not currently have the material science required to build a space elevator, but we can discuss the potential risks, consequences and benefits of building one based on known scientific facts.
All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Speculative or unreferenced answers, as well as those not supported by strong scientific theory, are not welcome. Long, comprehensive answers are desirable, but length and quality aren't always correlated.
In simpler term, reality-check is asking whether the setting is realistic within the given world (magic included). Realistic does not mean the same as scientific.
science-based indicates that the asker wants to find a solution to their problem based on Science as we know it today. So no magic, no handwavium, etc.
hard-science is a meta-tag. The asker is expecting a certain scientific quality in the provided answers. Whereas science-based, mentioning some scientific facts, common knowledge, etc. are ok, with hard-science, you need to be more specific in citing a given study.
Reality check is more about asking if what you already have makes sense, it can be science based but doesn't have to be (that is, you may be able to answer with common sense or some similarly unscientific method). Likewise, science based asks for answers that are based in science, but does not require you to have an already built scenario that you need checked.
Hard science is stronger than science based, it requires proof, either through sources or through a lot of math. Don't ask for this if you don't already know most of the math/science involved in your question. Similarly, if you use the hard science tag and an answer does not apply a lot of science with references or proofs, then you should downvote that answer (and explain why).