Esxi here. Got into virtualization with esxi and just stuck with it.
Also from what I've seen and haven't put much research into it hyper v requires a windows server to run on which means technically you would pay for 2(?) Licenses instead of 1 like you would with VMware esxi.
Again that's my impression and I could be wrong.
Hyper-V is a free standalone product, built-in to Windows 8 and above Professional & Enterprise SKU's, as well as Windows Server Standard and Datacenter. As such Hyper-V as a feature is as free as ESXi can be.
The real cost is management. In Hyper-V, you may or may not require System Center 2012 for Virtual Machine Manager depending on your management needs. ESXi requires vCenter long before Hyper-V (at least when used in Standard or Datacenter) requires SCVMM.
Frankly, if your server virtualization environment is running Microsoft workloads, then you are wasting money using anything than Hyper-V. The scale of waste depends on how many processors are being licensed for a third party virtualization platform. The more you have, the more you shouldn’t be using a third party virtualization vendor.
If you are running Windows VDI workloads, well, things are more muddled. Unidesk supports VMware now, and have been claiming for years that support is coming to Hyper-V. The layering in Unidesk for applications is superior to Application Virtualization. This makes Hyper-V a poorer option for VDI at this time because of Unidesk.
Personally, I wish Microsoft would fully license Unidesk for integration into System Center Configuration Manager and then connect ConfigMgr to SCVMM. If Microsoft were to implement it there, and then just enable the Windows bootloader to support virtual disk layering, then layering could become an excellent deployment platform for both VDI and physical workstations. (If you are not aware, the Windows 7 and above bootloaders support booting physical machines from virtual disks. Windows 7 can native boot from a VHD, the Windows 8 bootloader gained support for native booting VHDX, and the Windows 8.1 Update 1 bootloader gained support for native booting WIM files.)
Linux support for Hyper-V significantly improves with the newest kernel. Install, enable SSH, and your mostly good to go. Older versions may still require the drivers be installed, and some variants require a config file be changed to enable Hyper-V support. That said, I would personally never attempt to run a production Linux VDI environment from within Hyper-V at this time (if ever).