If you're looking for a managed solution where installation, upgrades and backups are handled by our administrators then choose the hosted version.
We highly recommend using the HOSTED version since it's being updated several times a day (so called "continuous deployment"), we also fine-tune our servers, our CDN-network, our database server, we keep tuning the heck out of our hardware - for instance, we have recently moved our helpdesk database "transaction logs" to an SSD drive to fasten performance.
We also make sure that the default email server that comes with the hosted version is using the fastest network, hard-drives, we fine-tune it to make sure all your emails are delivered (configuring reverse-DNS-entries, keeping an eye on the black-lists, etc. etc.). We have a dedicated super-fast Linux-based email-server that routes emails right into the helpdesk app, without any POP/IMAP middle-tier, via our own API. It makes sure your email convert into support tickets almost instantly.
It's not just Amazon. It's the fast-performance high-memory "instances", optimized for network storage performance and SSD local drives. Its the best Amazon can offer at the moment. It's daily S3-backups and hourly database backups. And even if (it happened only once over the last 3 years) the server crashes - a copy of your data is stored on a network drive, which is physically on a separate machine. More here.
Our hosted version supports Active Directory logins, single sign-on with apps that support SAML-protocol, and even has a customizable "authentication API" - if you want to control every tiny bit of it.
Cancel any time. We'll give you the data - a free backup of all your tickets. And if you ever decide to come back - we keep your data for 4 months, even for cancelled accounts, even the trial-accounts, that's the policy.
If you're looking for full control over your help desk software, backups and data then choose the downloaded version to install it in-house.
If your organization has a policy of storing your data in your own basement - we're cool with that, that's exactly why we have developed a "self-hosted" on-premise version. It's the exact same code-base and feature-set. Moreover - you get Windows-integrated authentication for your intrAnet users. If you have a Windows domain - all your users have to do is... open the app in their browsers.
Related article: why hosted helpdesk is better