As a student going into my senior year of college, I think this will be a huge weight off my peers' shoulders. There are countless job search sites that allow you to submit cover letters and resumes, but for each site you need to first upload the documents. This LinkedIn plug-in will eliminate making dozens of separate resumes for each job. Not to say that it is no longer necessary to tailor each application to each specific position; it will just be much easier now.
Once you click the Apply With LinkedIn button your profile snapshot appears with the option to edit. You can then create your profile for the specific position before clicking the button. And what would LinkedIn be without help of connections -- that is the whole point of the site! Once you have applied for the job a screen pops up listing your connections at the company receiving your profile. This feature makes it simpler to request recommendations, which may make the difference between employment or continued searching.
The plug-in will make profiles easily searchable compared to present methods. Each application will be in the same format so sifting through applications for specific qualifications with a few clicks of a mouse. This is a great feature since there is a large possibility for users to go apply-crazy and submit applications to jobs of which they are not qualified.
Overall, I think this is a great addition to the social media world. It is another example of how our world is changing. The almighty resume could be, in the far future, on its way out.