While this is a broad question, if someone is new to R this can be confusing and the distinction can get lost.
All data.tables are also data.frames. Loosely speaking, you can think of data.tables as data.frames with extra features.
data.frame is part of base R.
data.table is a package that extends data.frames. Two of its most notable features are speed and cleaner syntax.
However, that syntax sugar is different from the standard R syntax for data.frame while being hard for the untrained eye to distinguish at a glance. Therefore, if you read a code snippet and there is no other context to indicate you are working with data.tables and try to apply the code to a data.frame it may fail or produce unexpected results. (a clear giveaway that you are working with d.t's, besides the library/require call is the presence of the assignment operator := which is unique to d.t)
With all that being said, I think it is hard to actually appreciate the beauty of data.table without experiencing the shortcomings of data.frame. (for example, see the first 3 bullet points of @eddi's answer). In other words, I would very much suggest learning how to work with and manipulate data.frames first then move on to data.tables.
