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The lmerTest package provides an anova() function for linear mixed models with optionally Satterthwaite's (default) or Kenward-Roger's approximation of the degrees of freedom (df). What is the difference between these two approaches? When to choose which?

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I'm also interested in figuring out what the difference might be. The best I can offer you, for now, is that this blog post suggests that the Kenward-Roger approximation is slightly, but probably not significantly, more conservative than the Satterthwaite approximation. The author also notes that they are both more conservative than the normal approximation, but again, not by much if the sample size is high enough. I'm not sure whether or not this was a generalizable conclusion of the author's or not though.

Edit: I will add that the article "A comparison of denominator degrees of freedom approximation methods in the unbalanced two-way factorial mixed model" by K.B. Gregory seems to indicate that neither method is typically a better method, although there are apparently occasions where the Kenward-Roger approximation loses some level of conservativeness.

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Another difference between the two methods is described in Luke (2017):

Both the Kenward-Roger (Kenward & Roger, 1997) and Satterthwaite (1941) approaches are used to estimate denominator degrees of freedom for F statistics or degrees of freedom for t statistics. SAS PROC MIXED uses the Satterthwaite approximation (SAS Institute, 2008). While the Satterthwaite approximation can be applied to ML or REML models, the Kenward-Roger approximation is applied to REML models only.
(my bold)

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