What is the difference between Scrum Metrics and KPI?

Metrics obtained in Scrum are likely to be gauged at least once per Sprint, whereas KPIs cannot be expected to provide any such assurance.

The primary measure of progress is to have at least one Done finished Increment of working product each and every Sprint. Derivative metrics (such as velocity, the rate at which work is Done) ought to support this endeavor.

Scrum metrics like burndown chart, Average velocity, Cycle time, lead time, backlog ageing etc. are relevant for a scrum team to track progress and identify areas of improvement and are reviewed every sprint. 

KPI's could have a much wider scope taking into account how the team serves the business and how are we adding value to the customer. 

Can Scrum metrics be used as KPI's? - Answer would be a Yes or No based on the situation (probably scrum metrics wont show if you have built the right thing and which have a value for the end user). You may have to use NPS, Customer surveys, ROI etc which may be outside Scrum metrics.

Would like to hear from others too on this topic as it is an interesting question.

From kpi.org

Key Performance Indicators (KPIs) are the critical (key) quantifiable indicators of progress toward an intended result. KPIs provide a focus for strategic and operational improvement, create an analytical basis for decision making and help focus attention on what matters most.

Managing with the use of KPIs includes setting targets (the desired level of performance) and tracking progress against those targets.

Managing with KPIs often means working to improve performance using leading indicators, which are precursors of future success, that will later drive desired impacts indicated with lagging measures.

I am not including any definitive explanation of "Scrum Metrics" because none exist.  There is a multitude of information about possible Scrum Metrics but nothing definitive.  So ... 

What is the difference between Scrum Metrics and KPI?

One is something that has been defined while the other is something that is nebulous.  

Now, here is how I try to explain this to people I work with directly.  

I will use some specific metrics to help my teams see how effective they are.  These include flow metrics because they enable the team to see how work flows through their process by identifying bottlenecks, gauge effectiveness of procedural changes they make and understand their ability to complete work. I will use stakeholder feedback to identify how effectively the team is delivering usable updates that are seen to be valuable. 

I use Key Performance Indicators to help the organization understand how the teams are providing value and benefit to the organization.  For example, set a KPI of delivering usable body of work that is seen as valuable (think "increment") twice per Sprint. How often do we achieve that? That helps the organization understand how often the stakeholders are receiving the benefit of the team's work.

Any metric I use to help the team measure their own effectiveness (i.e. Scrum Metric) is solely for the team's information.  Any KPI that is used will be for the organizations benefit.  I also try to make sure that any KPI set at a team level will roll up to support divisional/corporate level indicators so that they are relevant to everyone.