The active install count is updated daily based on update checks from real active installs on active sites. Sites that do not receive any visit on a given day don’t count, neither do sites that have just installed the plugin and haven’t checked for updates yet.
The download count is always much higher than active install count because it is based on raw data. Each time someone clicks the download button, the count increases. Every update also count as a raw download, including on sites where the plugin is installed but not active.
It would be very easy to artificially increase the download count (just click the button like a mad man), but that won’t be of any use because the active install count – which is much more prominently displayed – will not be affected.
Otto is probably the best person to explain how this works, and his multiple comments in this thread provide at lot of additional insights: https://wpchat.com/t/wordpress-org-now-reports-active-plugin-installs/679/26