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.
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.