The primary difference between the two forms is the rhyme scheme. The Petrarchan sonnet has a rhyme scheme of ABBAABBA where the lines ending in "A" rhyme with each other, as do the lines ending in "B". The Shakespearean sonnet has an alternating rhyme scheme of ABABCDCDEFEFGG, ending with a rhyming couplet.
Both sonnet forms are always written in iambic pentameter, which consists of 10 syllables, in five pairs of stressed and unstressed syllables, in every line. Beyond this, and perhaps shared subject matter, the two differ greatly within their forms while still considered the same type of poem.
The specificity of the Petrarchan rhyme scheme, which was named for and invented by the Italian poet Petrarch, works very well with the Italian language as it has the ability to rhyme far more freely than is usually possible in English. The first eight lines, called the octave, presents an argument or question after which there is a turning point or volta. The question or argument is then answered by the sestet, the last six lines.
In comparison, the Shakesperean sonnet relies heavily on the final couplet which often expounds upon, refutes, or otherwise illuminates the first 10 lines of the sonnet.