Medium and Large site administrators are usually very similar. Many of the larger sites get highly siloed, but otherwise the tasks are generally the same. The focus is on preventative maintenance rather than firefighting. Small businesses tend to focus on firefighting. The larger the site generally the more automated functions get (or the more political, really depends on the business and industry).
You mention Engineer in the title. Engineers are generally people who specialize in system designs and installation. Administrators tend to do more long term maintenance. This is somewhat dependent on the business and region.
You may want to focus on automation tools and scripting if you're looking to move up. Some vendors are branding these tools as Cloud Management (though the same tools have been around long before management got preoccupied with the Cloud). Also business management, strategic planning, communication, and budgeting/accounting skills will all come in handy.