Translation:I will follow you through fire and water.
Late reply, but there's a long discussion here: https://forum.duolingo.com/comment/8262010
In short, they're mostly synonymous, but igenom is used in particle verbs, e.g. läsa igenom (read through), gå igenom (go through = review), tänka igenom (think through). According to my notes, each of these is used exactly once in this course (3 sentences total with igenom).