By big do you mean size of company or deployment size.
We used to do VMware but dumped it mainly because we got sick of the price tag associated with it. Innovation comes from employees not platforms. We use oVirt now in our corporate office for internal, dev and some production systems. We're in a hyper-converged hosted engine setup on CentOS 7 with storage (GlusterFS) running in replica 3 across 3 nodes on 10GBase-T switches.
Why oVirt and not vMware?Free is a big factor and should be considered in decision making process. The cost savings over 5 years by eliminating VMware licensing alone is huge. And the technical knowledge to manage oVirt in place of VMware is easily obtained. You can even try it FOR FREE!
Regardless of the cost however, oVirt is ENTERPRISE VIRTUALIZATION. It is the upstream project for Red Hat Enterprise Virtualization. In fact the docs for RHEV can be used to administrate oVirt (for the most part).
You get:
True web based management console (not that crappy Flash interface)
End User Portal
Active Directory integration
Memory Ballooning
Live Migration
Live Storage Migration
High Availability settings
QoS
Host Load Balancing
IPMI power management
VLAN tagging
External or Hyper-converged storage options.
VM resource allocation
Storage Geo Replication (onsite or across the internet to another site).
RHEV isn't going anywhere. And for there to be RHEV you need oVirt.
They've got some big names using them including British Airways.
Why do you use oVirt?
Again, it really boils down to licensing costs. VMware is not just an upfront cost. If you want to get updates you have to maintain your support license. oVirt does everything we need it to do, and more really, so purchasing VMware just doesn't make sense. We've also found VMware to not be as stable as oVirt. Our initial oVirt system has been running for about 3-4 years (I'd have to look at the ticket to get the exact date) without so much as a reboot. Our VMware systems kept getting kernel errors about every 2 months further prompting us to look elsewhere.