In general, Research mode is best for quick analysis and research, while campaign mode is best ongoing tracking.
Campaigns provide ongoing tracking on a daily or weekly basis for your tracked sites. Data can be much more specific for your site, since you can control which keywords are tracked, which part of your site is tracked, where it's tracked, and many other options. The number of reports and level of detail available is much greater than what's available in Research mode. Data may take between 1-24 hours to collect, and you're limited to the number of keywords you track.
Data in research mode is available instantly, but data is not updated as frequently. You're don't have any control over which keywords are tracked or where they're being tracked. There's no limit to how many keywords are tracked, but you're limited to the number of results you can pull from the database.