I'm not sure about Element but if the documentation says it isn't needed it likely isn't. I cannot say for sure (hopefully someone here will know better).
As far as the differences between Element OS and ONTAP, you have to keep in mind NetApp bought Solidfire so the platform is a secondary platfor, while FAS/ONTAP have been the legacy NetApp solution but modernized for modern IT/Cloud workloads. They have different use cases and both work well.
> Is it necessary to have Snapmirror licence on the SolidFire cluster as well?
It's not necessary.
As of 12.5, all SolidFire-based products (eSDS, SolidFire, NetApp HCI) still come with all-inclusive software licensing.
> Last but not least, what is the biggest difference between Netapp Element software and ONTAP?
SolidFire is built on a shared-nothing scale-out architecture that serves only iSCSI, mostly consumed in Virtual Infrastructure environments including containers (as stand-alone block storage, obviously) and suitable for small to medium individual workloads. It supports Win, Lin, VMware and just one storage protocol, so it's not a "unified" storage appliance like ONTAP.
Since you have the both products, you'll probably quickly form own opinion on how they compare. 🙂
By the way, there's a separate area for SolidFire-only questions (they've moved links around several times in recent years, I hope that link won't get broken soon).
It's OK to ask SnapMirror questions like yours here because it's pertinent to both platforms, but in the case you weren't aware of the other area and if you have questions related to only SolidFire, feel free to post them there.