What are the differences between teaching in the quarter and semester system?
There is another variable at play, actually. Some places 2:2 will mean two courses each semester where the courses meet 3 times (hours) per week, where 2:2:2 means two courses each "quarter", but where the course meet 4 or even 5 hours per week for fewer weeks. The students also, typically, then take fewer, but more intense, courses in a term. 

But that means that you may be able to handle about the same amount of material in a course under each system as you meet for more hours and the students are (hopefully) more engaged in your course. 

Another possible variable, is the quality of the students (and faculty). Dartmouth has traditionally been on the quarter system, and the students are very (very) good. And in CS, at least, they work very (very) hard. You may have a challenge keeping up with them. This is likely true for other similar institutions as well. 



I'll note for completeness that some institutions, including Dartmouth, use the quarter system so that they can run four terms of equal length per year (hence "quarter" system). Faculty normally get to choose three of those four terms, but may not, in general, be permitted to avoid summer quarter always. Students likewise, my be required to have schedules that include some summer quarters and some "time away" quarters (for travel in Europe, duh). This makes better use of other facilities and funds. 

Under such a system, it is also (theoretically) possible, if not encouraged, to have a year's "sabbatical" "every" four years if you teach for six consecutive quarters, then take off for a year and then teach six consecutive quarters. Twelve quarters over four years is a full load, if you are required to teach (on average) three per year. This requires the dean's ok, of course, which is granted only occasionally and for good reason.