Personally, my favorite depictions of psionics in fiction and RPGs often have these elements listed below. (The list is non-comprehensive, and no depiction has all of them. Some only have one or two.)
1) it is physically demanding, sometimes to the point that using your powers could render you unconscious or even kill you
2) it is internal to you; expression of psionics is a mastery of self. No outside conditions, materials, technology or beings are required to give you powers, but may increase your abilities or control
3) most of the time it’s not something just anyone can learn. You’re either born with psychic potential or you are not. Occasionally, a technology or “psychic surgery” can awaken psionic potential in those who are not so gifted, especially if the setting explicitly makes psionics genetic in origins, and features sophisticated DNA manipulation.
4) most psionics manifestors only have one or a small group of powers that they improve in with time and experience. Generalists who can “do it all” are rare.
5) psionics has limitations. Sometimes it’s general to a kind of powers, sometimes, it’s individual. Some limits are absolute. Others can be overcome by effort and/or training.
6) where magic and psionics exist, they are not the same. As such, each side has advantages and vulnerabilities against the other.