Microsoft Access 2013 is the software that has progressed over the ages from Access 1.1 in 1992 until now.
Office Professional 2013 Software (which includes Word, Excel, PowerPoint, Outlook and Access) will allow you to install and run Access 2013 as usual.
The Office 365 version is paid by monthly subscription and provides Web services to Office 2013. It includes the Office Professional 2013 software as well as Access 365 and Access 2013.
Access 365, besides the name "Access", has little to nothing in common with Access 2013 or previous versions of Access.
What is Access 365 all about?
Access 365 is a tool for creating web apps that run within SharePoint 2013. The old .ACCDB format has been abandoned. New applications should use the Access 365 model, or stick with Access 2010.
Access 365 allows the rapid creation of small applications using the standard web technologies of HTML5, CSS and JavaScript in a simplified web development environment. One big feature is that Access Services can now store data to Microsoft SQL Server or an SQL Azure database
The use of SQL Server overcomes the 2 Gigabyte size limitation of the previous Access databases. SQL Server also provides all the security and features essential for any strategic company business system.
And What about Access 2013?
The Access 2013 desktop version is much unchanged from previous versions - it still supports the Front-End and Back-End database model and Visual Basic for Applications (VBA).
But Access 2013 no longer supports:
The Microsoft Access project (ADP)
The Access 97.mdb database format
PivotCharts and PivotTables
The dBASE database
Smart Tags (Action Tags)
Access 2003 Toolbars and menus
The Microsoft SQL Server database Upsizing Wizard
The Developer Source Code Control