You are both right.  In one sense, it really isn't critical, but there are tangible differences that some folks find important - much like the way the frequency of changing strings is a personal preference.  Objectively, the surface of the rosin cake loses volatiles and becomes more glassy over time, losing some of its grip and becoming more dusty.  Some people periodically heat their rosin cake up to bring the oils within it up toward the surface.  Others just replace the cake every year.  Some, like me, use one for ages and never notice much difference.  As far as equipment goes, it's a pretty inexpensive way to give yourself a treat for $10 or so.  If nothing else, you could try a different type now and then.

The darker stuff tends to be grippier in cooler weather, while the blonde and amber versions work well in warmer weather.  There's probably a good compromise for your climate, but it's also down to personal preference.  The Hill dark is a basic standard, while Pirastro offers a whole range to match with their different strings.  I use their Oliv type, which is on the dark side, even though I don't play Oliv strings.  I like the grip.  I've also tried their Gold rosin, which is very smooth playing, more muted, and sweet.

I've been curious to try this stuff called Magic Rosin, but I think it's only available online.  It is colorless but still apparently very sticky.