The major difference between the two from the learning perspective is: the Jigsaw requires the students to work together to get a final project/answer and the competitive they work against each other trying to get the answer/final work before their peers. Typically the jigsaw is designed so that every one must do his/her part in order to accomplish a given feat. From a teaching perspective, both need careful planning and organizing to provide the optimum learning opportunity.
Both methods of teaching and learning have distinct settings and should be implemented when the student comfort level and teacher comfort level prove to be appropriate. The competitive learning style is where students are pitted against one another in the acquisition of learning and assessment, fighting for results. They understand that learning and success are contingent on they do in relation to others. Teachers can design this in terms of games and activities that create an atmosphere where competing against one another is an integral part of the learning process. Through this, one learns the value of healthy competition in assembling one's learning. In the jigsaw method of collaborating, each student is required to be an "expert" on a given sub topic in a larger topic. Each student does research, like an expert would, understands their component of the lesson, anticipates questions, and prepares to lead a mini lesson on their component. Once each student has prepared their own mini lesson, they share this with the other students, who in turn share their mini lessons. The jigsaw aspect is that each student holds a particular piece of the learning puzzle. When it is all assembled, a complete picture of knowledge about the topic emerges.
While previous two answers do give some useful information about jigsaw learning system, they fail to recognize and dispel a serious wrong assumption in the question. The question assumes that Jigsaw learning and competitive learning are two different varieties of competitive learning. Fact is that jigsaw learning system is a cooperative system of learning developed to replace the traditional competitive learning system, in which students, in their attempts for better grades, only compete with each other.
While the competitive system does create some motivation to learn, it also creates tension between students. This can act like fuel to fire in the tension that may exist between students because of their different backgrounds and personal likes and dislikes. To solve this problem, the jigsaw classroom techniques were developed in early 1970's.
In this system each student is given the responsibility to prepare one specific portion of the complete subject to be learnt. Thus if there are 25 students in a class of geography, each student will be assigned a specific topic within the total subject of Geography. Each student learns his portion by independent study and research under guidance of the teacher. Then each student, assisted by the teacher, presents his part in the class room. Thus each student is dependent on every other student in the class to learn the complete subject. This not only makes the atmosphere in the class more cooperative. The student acquire additional skills in self study, researching, and cooperative working.
Additional useful insights into jigsaw method, along with background information on development of this method, can be obtained from the Internet site referred below.
What hasn't been included in these answers is other purposes for both Jigsaw and Competitive strategies. The Jigsaw method is a great way to have students understand a great body of text without spending a great deal for time reading all of it. Furthermore it engenders the type of summary that many employers are looking for in that it asks students to summarize completely and consisely without adding their own opinions. Furthermore (althought this benefit has been mentioned) it helps develop student responsibility because they are aware that they must be responsible to the entire class.
For example in my classroom, I was teaching environmental education and had one copy of ONE WITH NINEVEH. I couldn't copy the entire book, nor entire chapters for each student, so instead I copied a part of a chapter for groups of students who read and made reports of the material. It was very helpful for individual students, groups and the whole class. Part of why it was helpful was that in working in groups with the same material, students were introduced to different reading strategies other than the ones on which they usually relied. Some students who didn't usually report, reported for the class. So again they learned new skills. Further many students learned to elaborate. And in terms of information lost, whatever wasn't explained in the reports, I later explained.
On the other hand competitvie activities don't always have to be each student competing against every other student. Teacher made teams are a good way to include competititon without engendering bad feelings.
Competition is great for memorizing multiplication tables or vocabulary words as long as everyone in class has the ability to memorize. Some times students can't memorize either because of a learning disability, stress, PTSD, etc. Then other means must be used to help those students to participate successfully with the others.