Looking at the HIA's website, Essential Starter is €90 cheaper than Aviva's Access Plan, but Aviva's plan will fully cover a private room in public hospitals, assuming you get one.
Essential Starter will fully cover certain types of major heart surgery in the Mater Private and the Beacon Hospital, but Access Plan will cover some of the cost of any treatment in any high tech hospital. This would still mean paying a substantial part of the bill yourself though.
Essential Starter's overseas cover is a little bit better but Access Plan covers more MRI, CT and PET scan centres. The maternity cover on both is minimum as well.
Ultimately, it depends on your specific circumstances, but I'd be inclined to go with Essential Starter. The cost is lower. Aviva's plan gives you access to more hospitals, but Essential Starter gives full cover in more hospitals. And Essential Starter's MRI/CT/PET scan cover would be adequate for where I'm living.