Probably worthwhile to augment all comments into an answer.
Both my comment and BenBolker's point to doc page ?Extract:
Under Recursive (list-like) objects:
Both "[[" and "$" select a single element of the list. The main difference is that "$" does not allow computed indices, whereas "[[" does. x$name is equivalent to x[["name", exact = FALSE]]. Also, the partial matching behavior of "[[" can be controlled using the exact argument.
Under Character indices:
Character indices can in some circumstances be partially matched (see ?pmatch) to the names or dimnames of the object being subsetted (but never for subassignment). Unlike S (Becker et al p. 358), R never uses partial matching when extracting by "[", and partial matching is not by default used by "[[" (see argument exact).
Thus the default behaviour is to use partial matching only when extracting from recursive objects (except environments) by "$". Even in that case, warnings can be switched on by options(warnPartialMatchDollar = TRUE).
Note, the manual has rich information, and make sure you fully digest them. I formatted the content, adding Stack Overflow threads behind where relevant.
Links provided by phiver's comment are worth reading in a long term.