They are several reasons why you might want to use different sized needles... some that come to mind...
A lot of knitters purl more loosely than they knit, and one way to compensate for that is to use one needle size for knit rows (in stockinette, this will be the right side), and a slightly smaller size for purl rows (wrong side).
Tight knitters might want to use circular needles with a smaller left needle tip so that their stitches are not as tight when knitting off on the left needle, but are the desired size when being formed on the right needle.
Amy also talks about using different sized needle tips when doing the 2 circ method for small diameter knitting on interchangeables. It's a way to avoid having to have 4 of the same size needle tips. (It uses the same principle of knitting off a smaller needle as described in the paragraph above.) See her video for details.
Hope that made sense. Given what you said about changing size between rows, I'd guess it's the first explanation, but I thought I'd mention the others too. Of course, it could be something entirely different.