I've been using these two recently so I thought I'd resurrect this thread and share my views.
As far as I know Sophos doesn't have a paid version, but Eset has.
Sophos first. In my opinion it is unquestionably the best free mobile antivirus. Unlike the majority of free apps, it has web protection, and is also the lightest of all similarly featured free android AV's I've tried, although not a light as Eset, but a lot lighter than the resource heavy Avast, which, unlike Sophos, also has rather annoying ads. I also found Sophos much lighter than BitDefender, and very different to the figures in this test:
https://www.geckoandfly.com/17030/android-antivirus-malware-security-app/
Virus definitions are updated automatically but one slight criticism I have is they don't appear to be updated at all over the weekend, even if you try to do so manually.
Eset free is the lightest of all the android AV's I've tried, but unfortunately, unlike the premium version, it has no web protection and it doesn't update automatically.
So to conclude, if it has to be free, Sophos is the clear winner. A little heavier than Eset free but fully featured. Right now I am using Eset premium simply because I have some free keys. Otherwise it would unquestionably be Sophos.