In short, the headlines are: All three licences allow access to CRM from all available clients eg browser, outlook, tablet, phone. (This is a big difference from the 2011 ESS CAL).
All three licences allow read access to all data, including custom entities. (subject to Security Roles of course).
Essential allows read/write access to activities, activity feeds and custom entities.
Basic allows everything in Essential, plus read/write to Accounts, Contacts, Cases, Leads. Also access to reports, and to create personal charts and dashboards.
Pro allows everything. Most notably Sales and Marketing (Opportunity, Quote etc, Campaign, Marketing List), plus service management (facility/equipment and all that).
Pro is needed to build customisations, but not to use them, which is where I think some confusion has arisen.