Does Cisco ACI use VXLAN inside the fabric or is something else used instead of VXLAN?
ACI uses VXLAN but not in a way that would be (AFAIK) interoperable with any non-Cisco product. While they do use some proprietary tagging bits, the real challenge is the control plane.
NSX does use VXLAN but has features beyond ACI such as load balancers and firewalls.
Firewalls - yes, including distributed ones. Load balancers - yes, Edge Services Router can also work as a load balancer. Distributed load balancers? Works best in PPT; it's an awfully hard task to solve.
I understand VXLAN is a standards based protocol and NSX and ACI are vendor terms but how do they inter-relate?
VXLAN is just a data-plane protocol (like IP or Ethernet). The problem is the control plane - how do you figure out where everyone is (think OSPF or BGP). I think the intersection of NSX and ACI control plane protocols is still zero.
I heard recently in a VMware session saying you could use NSX with ACI and I scratched my head and thought why you would do this, what would this provide over the added support burden?
Sure you can. You build an ACI fabric to provide IP transport and run NSX on top of it ;) A bit expensive if you ask me ;))
However, I know production deployments running VMware NSX on top of Nexus 9000 switches using EVPN control plane. Mitja Robas talked about his experiences with one of those deployments in the last session of Autumn 2016 Building Next-Generation Data Centers online course (and of course you’ll get access to the recording of his presentation if you register for the Spring 2017 session).