There wasn't really a difference between them, they held the same powers. The senior consul was the one elected first, according to Cicero, which also meant he got more votes because of the way that elections within the comitia centuriata shut down voting once a majority was reached. The only real difference between them was in deciding who presided over the senate in each given month (the senior consul took January as I recall, and had every other month) and as I recall Cicero mentions some practices regarding the order of consular speakers within the senate, but I'd have to slog back through his speeches to find it. In practice, the senior consul was usually, though not always, the candidate with more influence (which isn't surprising, if he's the guy who gets more of the vote). The only other difference I can think of is that during Marius' consulship his senior consul led the first consular army and Marius only raised a force as junior consul when the senior consular army proved inadequate to the threats Rome was facing elsewhere

She may be referring to suffect consuls. These were elected generally after the death of a consul as a replacement in the Republican era. They became very common after the establishment of the Principate once consulships started being handed out as prizes for those in Imperial favour - you might seen two or three cos. suf. in the rolls on any given year. The first consul of the year was the senior with most prestige, diminishing as more were chosen.