First of all SharePoint Server gives you many more possibilities. In SharePoint Online (SPO) environment you are sharing a server with several other tenants (other subscribers) and have limited access to multiple functionalities, e.g.

• Powershell cmdlets: SP on-prem vs SPO
• View Reports
• Features

Here you can find Microsoft's comparison of the websites. Bear in mind that SPO public websites will be discontinued and are not available in new tenants already.

On the other hand, with SPO you don't have to worry about the infrastructure, hardware or expansion costs (you just buy another license). There's full support and they promise 99,9% uptime

There is a third option: best of both worlds with a SharePoint Hybrid, which sounds like something you are going to make:
http://technet.microsoft.com/en-us/library/jj838715(v=office.15).aspx
http://technet.microsoft.com/en-us/library/dn607308(v=office.15).aspx

Have a look also at another Stack Question: SharePoint Online or SharePoint Server 2013

First of all SharePoint Server gives you many more possibilities. In SharePoint Online (SPO) environment you are sharing a server with several other tenants (other subscribers) and have limited access to multiple functionalities, e.g.

• Powershell cmdlets: SP on-prem vs SPO
• View Reports
• Features

Here you can find Microsoft's comparison of the websites. Bear in mind that SPO public websites will be discontinued and are not available in new tenants already.

On the other hand, with SPO you don't have to worry about the infrastructure, hardware or expansion costs (you just buy another license). There's full support and they promise 99,9% uptime

There is a third option: best of both worlds with a SharePoint Hybrid, which sounds like something you are going to make:
http://technet.microsoft.com/en-us/library/jj838715(v=office.15).aspx
http://technet.microsoft.com/en-us/library/dn607308(v=office.15).aspx

Have a look also at another Stack Question: SharePoint Online or SharePoint Server 2013