Summary involves restating every main idea from the original piece while also paying attention to the proportion of the original. That is, if one third of the original essay was devoted to a particular topic, then one third of one's summary should also be devoted to that same topic. When summarizing, one must capture every main idea presented from the original but do it in a fraction of the original space. That's why summary is so useful.
Paraphrase, on the other hand, involves restating the original piece word for word but using fresh language. For this reason, paraphrases will almost always be a little longer than the original. This is not always as easy to do as it may sound! Paraphrase has much more limited usage altogether. One would not attempt to paraphrase anything much more than a paragraph or a handful of lines, for example. I have used paraphrase in the past -- as an instructor -- when I was teaching Shakespeare's Hamlet, for example. Hamlet's soliloquies -- like the famous "To be or not to be" speech -- are very difficult if one is not familiar with the type of English Shakespeare used. For that reason, I have often made a word for word "translation" of some of Hamlet's soliloquies so that students can better understand them. Another use of paraphrase is to explain a very dense and difficult passage of prose writing, such as a technical explanation or philosophical passage, for example. For students like yourselves, you might find it helpful to take a difficult short quotation and rephrase it (paraphrase it) in order that your readers will understand it more easily.