Hana Cloud Public platform:
This is a public cloud providing on the market through SAP as well as some other corporations (e.g. Amazon.). You can simply go on the sites, create your account and begin with a recent instance of Hana.
1. Quick, easy set up in minutes
2. Low monthly costs
3. Great for standalone applications/development
5. numerous choices out there - Infrastructure only, infra + License, PaaS, third party providers etc.
6. Good for trials
1. These are primarily for standalone applications.
2. SAP doesn't directly assist you with connecting the Hana instance along with your on-premise systems, does not facilitate with any data loads etc.
3. Out of your company's firewall/network.
4. Not easy integration with your organization's systems (e.g. SSO, Security, etc)
SAP Hana Enterprise Cloud:
This is an Enterprise SAP HANA managed cloud providing. Includes Infrastructure + Managed Services on a monthly subscription.
1. SAP support with setting up and connect with your organization
2. although it's cloud, SAP provides a dedicated server for your organization which can be custom-built as required
3. Can be configured to act as if the inside firewall
4. Push / Pull possible (depending on security requirements)
5. Customizable hardware
1. Costly (by an order of magnitude)
2. Not really a 'cloud' (more like IaaS)
3. More time required to set up
Coming back to your question - for your mission-critical application - the selection can depend upon your necessities, who is the end-user / customer of the application and integration required with different systems (mission-critical and standalone do not go well along in my experience). It will also depend upon the sort of accessibility you wish for your end-users, kind of authentication mechanism and turnaround.
Just going by the word 'Mission Critical', I might go with SAP HEC.
For more related questions like this, you can go through these SAP HANA interview questions.