What is the biggest difference between HPE SimpliVity and VMware vSAN?

There are a lot of technical differences between both solutions but there are a few notes that are key to understanding the differences.

1. SimpliVity is not tied to a single hypervisor; rather it can work with KVM, Hyper-V and VMware hypervisors.
2. VMware vSAN is kernel-integrated meaning it runs at a much lower level in the processor stack than SimpliVty. Depending on your workloads this may provide a small boost in performance.
3. To combat #2, SimpliVity uses a hardware accelerator in its systems that accelerates writes by acting as a cache and provides compression and deduplication acceleration. In some workloads, this may provide a small boost in performance.
4. While SimpliVity is only available on HPE servers since the acquisition, vSAN works on literally any x86 server so long as your hardware (most importantly drives and HBA's... avoid RAID controllers in hardware) is on the HCL. If you're a current HPE customer then this might not be an issue, but if you are an IBM, Dell, Cisco or Supermicro customer this might be a problem.
5. As an adjunct to #4, there are certified platforms from all the major manufacturers for vSAN; so-called "Ready Nodes" which provide a full stack of hardware that's on the vSAN HCL.
6. I've found support to be a mixed bag. While HPE will do some support of the hypervisor, it can become a "blaming" match between HPE and the hypervisor vendor when issues occur at that level. With vSAN you only have VMware to deal with. However, in fairness, I have found VMware to be very quick to blame hardware which can lead to some finger-pointing as well.
7. vSAN is also available as a "full-stack appliance" like Simplivity where a single vendor provides and supports everything in the stack; that being VxRail from Dell.
8. vSAN's integration into vCenter is obviously second-to-none in terms of storage management. Third-party plugins to vCenter are good but you get into the "two-software-vendors" duel again when calling for support.

General advice though, make sure to analyze your workloads properly before proceeding with any HCI solution. There are salespeople from a number of big companies, HPE and Dell in particular who will tell you that their HCI solution is the answer to ALL your problems. This is patently not true. Many workloads won't fit in HCI due to hardware, software or performance restrictions. HCI can be very cost-effective for general-purpose workloads for office type environments but lack the peak performance and custom components of a solidly-designed 3-2-1 solution (server, switch, SAN).

It's also worth noting that the sadly ignored solution of up to 4 servers attached to a shared SAS array of disks can still provide an incredibly low-cost yet highly performant VM cluster using your hypervisor of choice without having the extra layer of software or management for the storage, a SAS array is usually a set-it-and-forget-it setup. It lacks in scalability but for small deployments can be much more cost-effective than an HCI solution particularly one where you know you'll never grow it (remote offices for example).

I would say the biggest difference is that SimpliVity is a Hardware and software HCI solution and vSAN is a software-only HCI solution. That being said, all the technical differences are related to those approaches. Each solution has its own advantages, for example, SimpliVity is really good with data optimization, compression and deduplication, for a big space workload you would probably need fewer nodes than vSAN, in the other hand vSAN has a unique integration with all VMware software suite, so if you are planning to do more advanced use of your platform you will probably have a more complete suite with the whole stack. Actually, when you plan to go for an HCI solution you should be looking for something else on top of just virtualizing storage workload, so that could be a good point between then. For those who would say that VMware will lock you up with the whole stack, HPE will do the same with SimpliVity, so you don´t have to look who is not going to tie you up, instead you should look for who has the better approach and what terms are the ones that I would like to be tied up and what I don´t.

The inline compression engine of HPE is very powerful (probably best in the industry). This makes backup and retention a simple and easy job. One of our clients was able to squeeze a nightly that ran 5+ hours into less than 60 mins.

The simple answer is that a five year TCO analysis helped break the tie between two very good HCI solutions.