If you choose AWS Elastic because you don't want administration overhead, I think it's not correct way to go. Actually the products are a little different there's an updated list of difference you can check here to get better opinion before making your choice.
Also you can check the forum about problem occurred with AWS elastic product to know if you will be affected or not. Mainly are old or already fixed, but as they are always a version behind, depends on your needs it can be cumbersome. If you still consider using AWS Elasticsearch, I think migration to elastic cloud can be done easily.
About the cost you have a list here and you even have a free trial. I highly suggest you to try and after that it will be hard to make another choice.
Just for information: If you use it only for logs management, and you manage your own servers it's quite easy, less than 1 day to upgrade to new version and apply the configuration changes, but using the elastic cloud may take a lot less (I manage my own servers).
And one last point that you need to consider is the support, this forum or slack channel are really active and there's a good community.
Disclaimer: I'm not working at elastic and I don't want to denigrate AWS Service I just share my opinion.