The biggest difference is that arrays are all rationally designed and therefore are biased for known transcripts and transcript isoforms. Sequencing libraries do not have this bias and so you should be able to detect all transcripts in your sample.
Another way to put this, which is indeed the biggest advantage: there is no need for a reference genome or transcriptome.
The biggest is what Micheal posted in his reply: You have a much greater dynamic range with something like RNA-Seq compared to a Microarray. You do still have problems, especially with trying to find significant signal, when dealing with very lowly or highly expressed samples, but it is still far superior to an array in this regard. And as Matt pointed out there is less bias as well, since arrays are probe-based. You can see "novel" transcripts and isoforms in an RNA-Seq based experiment.
Depending on experimental design and library you can interrogate both mRNA, and non-coding RNA as well. Another big advantage.
Recent paper addressing this question in depth: http://www.plosone.org/article/info:doi%2F10.1371%2Fjournal.pone.0078644
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higher dynamic range and sequence information