They are essentially the same, the later calling the former.
However tf.contrib.fully_connected adds a few functionalities on top of dense, in particular the possibility to pass a normalization and an activation in the parameters, à la Keras. As noted by @wordforthewise, mind that the later defaults to tf.nn.relu.
More generally, the TF API proposes (and mixes somewhat confusingly) low- and hi-level APIs; more on that here.