I don't think that a generic answer would be valid for every European continental country as University careers are already quite different from country to country.
As you are mentioning CSIC I am going to focus in Spain.
Short answer: Making your career at CSIC or any other Spanish research institution means that you have the opportunity of focusing on your research in a high quality institution. It also means that the possibility of changing to a University (again in Spain) decrease the longer you stay there, virtually becoming zero after some five years
Explanation: Spanish academia requires that you go through a system of pre-qualification (acreditación) for any of the four current faculty positions. Other countries such as France (Qualification) and Italy also have a similar system but I believe that the particularity of the Spanish case is that you need to fulfil a lot of different requirements in teaching, research, knowledge transfer, etc. You can check the requirements in this link (sorry, no English version). Being able to fulfil these requirements is relatively straight-forward for the first faculty position (profesor ayudante doctor) but it becomes increasingly difficult for the next ones if you are not employed at a University. It has been often quoted that a Nobel prize not previously employed at University would not be able to become tenured in Spain regardless of the quality of their research.
As you mention being currently in a research + teaching position at the UK, your previous teaching + supervising experience might allow you to become qualified (acreditado) for faculty positions. But both Spanish universities and the organism that evaluates research (ANECA) are extremely picky with the certificates that you need to bring from your home university. It is possible that some are deemed not valid because they are not signed and stamped by the right person, etc.