Note here: throughout I refer to imputation and mean specifically multiple imputation. Any other form is biased and incorrect.
Imputation is actually a procedure that makes a lot of sense. I would be in the camp to recommend doing it always. However, it is often not performed because it is rarely the case that missingness is so prevalent that a complete case analysis fails to obtain significant findings whereas an imputed analysis would. The assumptions are easy to state, but there are no tests to assess them: so they must be taken with a grain of salt.
Imputation gives a power boost to your analyses. That boost is proportional to a few, somewhat complicated considerations. Imputation works best when many variables are missing in small proportions such that a complete case analysis might render 60-30% completeness, but each variable is perhaps only missing 10% of its values. The power boost is much less impressive when one important variable is missing in 70% of cases, because the uncertainty in estimates will yield highly varying imputed datasets.
Imputation usually doesn't introduce many additional assumptions in most analyses, and can be quite robust. In longitudinal studies, however, one must take care to assess data for informative missingness. This is not a statistical consideration in the sense that tests may be applied, or in which sensitivity analyses can be conducted: this is simply a practical consideration.
Informative missingness is when inclusion probability or loss-to-follow-up depends on unmeasured factors in the analysis. This is especially the case when dealing with high risk populations and high risk behavior, such as drug use, suicidality, or even education (children who dropout are likely to have dropped out because of poor performances). The effect is that of unbalanced design and non-representativeness. The effect is somewhat similar to unmeasured correlation in linear regression: it doesn't bias your analyses, it just gives incorrect inference. That can go either way, really. It's totally conceivable that a complete case analysis gives significant results and the imputed data (despite having more observations) is not significant--and the imputed inference is correct.
Imputation technically isn't necessary when using fully parametric methods such as Bayesian regression or maximum likelihood. This is somewhat different, but if you write out the full likelihood integrating over the missing values, you'll see that the full likelihood is proportional to the complete case likelihood, so they have the same maximum. In my thesis (unpublished) we did show that, despite this, you get power boosts from imputation. So there are reasons to still do this.
ref: statistical analysis with missing data, Little & Rubin 2002 2nd ed