How is the career path of a full-time Research Associate different from that of a tenure-track / full-time professor?
Tenure in the US, and I assume most other places, implies a lifetime commitment to employment. You can't be fired for any but egregious misbehavior. The research associate probably doesn't get that guarantee and may work on a fixed term (but renewable) contract.

The very idea of tenure is to make it possible for tenured faculty to explore unconventional and even controversial ideas without fear of retaliation. Sometimes that is abused, but mostly it seems to work. Academic Freedom is based on tenure. 

Tenured faculty have different responsibilities than (only) research. Most teach. Most advise graduate, and maybe undergraduate, students. Most have committee responsibilities - for the governance of the department and the university. 

When General Eisenhower was made president of Columbia University it is said that in his inaugural speech he spoke at length about all the great things "the University" would do for "the Faculty". After he finished, one of the top faculty members, I don't know who, perhaps a dean, rose and said something like "With all due respect, General, the University can't 'do' anything for the Faculty. The Faculty IS the University." That is an old tradition - the governance of the university lies in its faculty. It is going out of style nowadays, but remains a dearly held tradition. Tenured faculty (some at least) take that responsibility very seriously.  

The research of tenured faculty and that of research assistants (or associates or ...) are likely funded in the same way. That would mean grants at most large places and internal funding, perhaps, at small colleges though small places don't typically have pure research positions.

It may be that the transition of the person you "saw" was caused by information that he/she was unlikely to achieve tenure and took the research option. Tenure is far from automatic and in many places is very difficult to get and lacks rationality. Some are "denied" tenure for no better reason than that the university is reducing the size of some department due to falling student enrollments. A person can even get a positive vote/recommendation from the tenure committee and department but be denied tenure for financial reasons. 

On the other hand, the person may have decided that the non-research responsibilities of tenured faculty simply weren't appealing and wants to devote him/herself solely to research, though without the lifetime employment guarantee. 

Complicating the decision is the fact that in most places, if you actually apply for tenure, go through the process, but are denied, you are usually asked to leave. There may be exceptions for some, but that is pretty typical. So, reading the handwriting on the wall may be wise for anyone who is borderline in any way or in a situation in which tenure is unlikely for any reason.