Pearson's $r$ is a linearity index that quantifies how well two variables $x$ and $y$ are related by the following equation: $y=mx+b$. In contrast, the consistency ICC is an additivity index that quantifies how well $x$ and $y$ are related by $y=x+b$. Finally, the agreement ICC is an agreement index that quantifies how well $x$ and $y$ are related by $y=x$ (see McGraw & Wong, 1996).
If you apply these measures to inter-rater reliability as in your question, then $x$ and $y$ are two different raters' ratings of the same objects. So Pearson's $r$ will allow the raters to have their own means (with difference $b$) and for one rater's ratings to be multiplicatively higher or lower than the other rater's ratings (by a factor of $m$). Consistency ICC will allow the raters to have their own means but not to be multiplicatively different. Finally, agreement ICC will require the raters to provide the same ratings.
The choice of which measure to use depends on which behavior seems most reasonable for your purposes. If you are okay with the raters having their own means, then the consistency ICC is fine. But if you truly want the raters to be interchangeable, then the agreement ICC is probably best.
McGraw, K. O., & Wong, S. P. (1996). Forming inferences about some intraclass correlation coefficients. Psychological Methods, 1(1), 30–46.
