Just as nauseous and nauseated use the same root word but have entirely different meanings, the same can be said about systematic and systemic. They both have “system” as the root word, but systematic and systemic do not have the same meaning and cannot be used in the same way.
If something is done systematically, then it means that it is being done in a methodical fashion. There is a step-by-step procedure being followed. This would be the case with an exercise regimen, which is a systematic plan for fitness. There exercises inside of this regimen are working within the confines of an overall system to achieve an overall goal.
You could say that the books at the library are organized in a systematic way: the Dewey Decimal System. If you go to one library and find a certain genre of books with call numbers in the 800-850 range, you’ll find the same genre of books classified with the same 800-850 call numbers at another library. The Dewey Decimal System, by definition, is a system. Thus, the books are organized systematically.
The word systemic, on the other hand, is used in an entirely different way. It means “belonging to or affect the system as a whole.” For example, some of the pesticides you may use in your garden are referred to as systemic pesticides. This is because they are designed to kill the pests by invading their systems and working their way through their tissues. There are systemic diseases too, which affect the whole body rather than one part or another.
Some people may say that there are systemic issues with the modern day banking system. Because of the way that banks are run, there are certain problems that exist and the only way to fix these problems is to overhaul the system. Similarly, some people may refer to the systemic misogyny that exists in certain cultures. The poor treatment of women is ingrained in the very nature of the culture and it is only through broad-sweeping changes that this systemic issue can be properly addressed.
For most situations and casual conversations, it is more likely that you want to use “systematic” rather than “systemic.” This can be a tricky difference to understand, but you will find that “systemic” is usually only used in the context of science and politics.