Careers at the National Research Council of Italy and difference from the university system
First, CNR is a giant entity with a hundred units, spread across all Italy, and with about eight-nine thousand researchers. It is therefore quite different from a university.

Yes, there are public competitions. All available calls can be found here. These calls are also published, like all public calls in Italy for whatever position, in the Gazzetta Ufficiale 4ª Serie Speciale - Concorsi ed Esami (in Italian).
In principle, technologist and researcher are two different paths. A technologist is a kind of senior technician. However, the distinction between the two paths is frequently fuzzy. For researchers, the career progression is Researcher (equivalent to an Assistant Professor in the university system), Senior Researcher (equivalent to an Associate Professor), and Research Director (equivalent to a Full Professor). For technologists, the career progression is Technologist, Senior Technologist and Technologist Director. You move from one level to another through competitions. These are all permanent positions.
CNR staff does not have any teaching duties, because it's a research only institution. Some researchers, though, are also affiliated to universities (and some CNR units are hosted by universities) and may teach classes. The salary between CNR and universities is about the same. Within CNR, also the salary between technologists and researchers, at equal level, is about the same. You can find details about the gross salary in these tables, which are however quite hard to interpret. The salaries for researchers can be found from p. 3 of the linked document. According to my understanding, the levels (abbreviated Liv. in the document) correspond to the positions: Liv III, Researcher; Liv II, Senior Researcher; Liv I, Research Director. The fascia corresponds to the number of years of employment: one starts from 1ª fascia, then after three years moves to 2ª fascia, after five more years to 3ª fascia and so on (the number of years between each fascia is reported in this page).