FIP actually focuses more on pitcher only performance in the context of what the pitcher can actually control himself, hence “Fielding Independent.” The inputs are HRs, Ks, and BBs (HBP too). It’s a three true outcome stat. Defense doesn’t come into play like it does with ERA+. FIP is more stable in that it doesn’t rely on defense or the luck of a hard hit ball becoming an out with no runs or a weak blooper driving in two runs. So FIP attempts to take out that “noise” to measure a pitcher’s performance. That’s why it is more stable over the course of a season or several seasons and why it’s a better predictor of what a pitcher will do next month or next season. ERA+ on the other hand, while it is a pitcher stat, is really a team defensive stat.
I guess a way to summarize it is a pitcher pitching one season for a good defensive team like the Cardinals and then the next season for a bad defensive team like the Nationals would be expected to have a similar FIP with both teams, but his ERA+ would likely be better with the Cardinals.