final say on order product backlog

#1 - The Product Owner has the last say, and is responsible for representing the interests of the customer to the organization.

#2 - It is my opinion that there is no difference, although prioritizing implies ranking based upon business value and other factors, while ordering may imply just moving things around.

It could be said that the ordering of a Product Backlog takes into account many things, including priority, technical risk, and dependency. Both the Development Team and the Product Owner ought to participate in Product Backlog refinement so these matters can be considered.

Since the Product Owner is solely responsible I’d say he’s got the final say as he’s the only one managing the Product Backlog 😊

As @Ian mentions ordering takes many things into account.  I'll give a example situation

Item A is estimated to 10 weeks to complete, provides value to 33% of the customer base, and could result in $500,000 in revenue.

Item B is estimated to 2 weeks to complete, provides value to 98% of the customer base and provides solution that has been requested by the customer base for 6 months. 

Item A could be argued to be prioritized as the #1 item due to the potential revenue.  However Item B could be ordered as the #1 item due to the shorter duration to provide enormous good will with the majority of the customer base. 

The PO is the one ultimately responsible for the ordering since they know most about the stakeholders requests, the business benefit of the item, the end user benefit, the ability of the stakeholders to absorb a release. However ordering should be done by using all available information and any source of information should be tapped.