Should I use missing data imputation with a model that already allows incomplete data? I'm just starting to learn about missing data imputation methods, and I'm confused. In every introduction I've read, the author starts by describing listwise deletion and says that it's a bad idea because it reduces your N. They then explain methods you can use to fill in missing values so you can do your analysis with all your participants. That makes it sound like the purpose of imputation is to avoid throwing out other data points that you did observe.
Question 1: Is imputation useful only because it lets you use the values you observed? Or do the imputed values themselves improve the analysis? 
For example, let's say I'm analyzing a repeated-measures dataset with a linear mixed model, and some participants are missing some timepoints. (Let's also say that the data are missing at random). Linear mixed models already work with incomplete data. Is doing imputation on the missing data points still appropriate, even though their missingness isn't causing me to throw out any other data?
Question 2: What happens if I use full information maximum likelihood to impute values for the missing data points? It seems like that does provide some additional information for my model -- I get to include the portion of the missing data points that can be inferred from other variables in my dataset, even if those variables aren't part of the model I'm testing. But is that actually legitimate?
 A: Question 1: in the situation you describe, if the repeated measures satisfy the MAR assumption, given the variables used in the repeated measures model, estimates will be unbiased (assuming you are using ML/REML for estimation). If you were to multiply impute the  missing values, using a model which makes the same assumptions as the repeated measures model, you will get back estimates which are pretty close to those from just applying the repeated measures model to the observed data.
In this setting, imputation can still be useful in certain situations. For example, you may have a variable which if you were to condition on it would make the MAR assumption more plausible, but you may not want it in your analysis model. In this case you can condition/adjust for it in the imputation model but leave it out in your repeated measures analysis model. Another situation where imputation might be useful in this context would be if you had missing covariate values, which the repeated measures model won't handle for you.
Question 2: if you use ML/REML to fit your repeated measures model, one can view the procedure as implicitly imputing the missing outcomes, although the procedure used to find the estimates does not need to actually impute them. Nevertheless, as per my answer above, no information/precision/efficiency is gained.
In a simpler situation that this true is perhaps easier to see: suppose Y has some missing values, and your analysis model is the regression of Y on X (as covariate). Then you could impute the missing values in Y (multiple times), then estimate the regression of Y on X using the full n, and combine estimates with Rubin's rules. However you will get back estimates (of the parameters in the model for Y|X) which are no more efficient than the complete case estimates. This is because although the incomplete cases give information about the marginal distribution, they do not give any information about the conditional distribution Y|X of interest.
