Custom Portal:-A Salesforce Customer Portal provides an online support channel for your customers—allowing them to resolve their inquiries without contacting a customer service representative. With a Customer Portal, you can customize and deliver a visually stunning user interface to your customers, and use the following Salesforce features to help you and your customers succeed.
Use Of Custom Portal:-The Salesforce customer portal is generally geared toward providing you a way to allow your customer's to see information that relates only to their account with you; things like their Account and Contact information, maybe the contracts they have on file with you, their Assets, etc. The primary limitation of a customer portal is that the portal user can only view their own account information.
Partner Portal:-A partner portal allows partner users to log in to Salesforce through a separate website than your non-partner users. Partner users can only see the data that you have made available to them. Furthermore, partner users can only manipulate data if you give them the appropriate user permissions.
Use Of Partner Portal:-The Partner Portal is used to allow access to external organizations to your salesforce.com database, typically for leads and opportunities. Think of it as a Customer Portal for use by external organizations (such as marketing organizations, vendors, third-party warehouse and shipping employees, etc). It does not relate two organizations together (that would be Salesforce to Salesforce), nor are you becoming "salesforce.com partners", either. It's used for your partners to access data that your organization allows access to. From a business perspective, it's far less expensive to grant them portal licenses than to create full-blown salesforce.com-licensed users.
Differences between Customer and Partner Portals:-
With each passing release the differences between the customer and partner portal, from a functional perspective, decrease. I expect at some point they will come together.

The main functional difference is that the partner portal has leads and opportunities so you generally (but not exclusively) roll it out to your selling partners where they can track leads, register deals, manage opportunities. etc. They both expose cases, content, custom objects, support Visualforce, have a similar user set up process, etc.

From a licensing perspective I suggest you contact your Salesforce rep for pricing but the partner portal is more expensive than the customer portal given that is supports the sales process.


Portals are replaced by communities from 13th release.



If you want to share data with your partners then Communities is now the way to go as Partner Portal has been retired as of the Summer '13 Release:

Starting in Summer ’13, the partner portal is no longer available for organizations that aren’t currently using it. Existing organizations continue to have full access. If you don’t have a partner portal, but want to easily share records and information with your partners, try Communities. Existing organizations using partner portals may continue to use their partner portals or transition to Communities. Contact your Salesforce Account Executive for more information.



Regards,
Subramani.J
Trinay technology Solutions