Lightroom and DPP are both built around a Raw processor, and if you have one you don't "need" the other -- what you will want is to add an image editor such as Photoshop/Elements to do editing tasks that go beyond Raw processing.
As to the differences between DPP and Lightroom, well, each has advantages. Lightroom has an extensive set of tools that make both developing and organizing images go quite smoothly and efficiently, as well as outputting images for various things. And, it has great tools for handling volumes of photos and, say, applying various settings to "batches" and such. So LR is a top-of-the-line app for a lot of things, and therefore comes at a price.
DPP has some "star" features, in that it "reads" and applies your in-camera settings (such as Picture Styles) to a Raw file Preview, and it has tools that match those in your camera so that you have a familiar "playing field" to work with. And, as a free app, DPP can actually do a complete job of processing and converting a Raw file for final use, so that makes "free" a pretty significant value
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As to which is best for you, I'd say install DPP and get to using it and get familiar with its various features. Then, once you are comfortable with DPP, check out Lightroom -- start with reading up on it, including the Lightroom 3 Help pdf that you can find online, and, when you are ready, download and install LR as a Free Trial -- you then have 30 days to explore! Continue reading up, and then you have many online resources where you can follow tutorials and such.
After 30 days, you should have a good idea as to whether Lightroom is worth the price, or if DPP is plenty to meet your needs!