Cloud version has limited capability, you can't do everything you can do in server version. You will have nearly all of core JIRA capabilities but can't install most of the plugins on the marketplace. But it is cheaper and maintained for you. You don't have to deal with server maintenance, backup. You always get the latest version of JIRA, you always get latest security fixes immediately.
It a give and take relationship. Some things only exist on the standalone, and some plugins are only available on cloud. The main reason for this is that the cloud version is somewhat higher than the standalone version, which means the plugins are available on the market place, but not compatible with the latest standalone versions.
The main reasons I see people choosing Server over Cloud are:
Connect vs Type-2 addons. Cloud has a short, fixed list of type-2 addons you can use, Server lets you install any of them. Connect addons are only available on Cloud. Type-2 addons are vastly more powerful and flexible than Connect ones (but more complex to write, maintain and quite easy to get wrong if you write your own)
Scale - as Oliver mentions, there's a storage limit to Cloud. There's also a user limit, and people wanting thousands of users must use Server (and even its big brother Data Centre for really big installs)
Speed - for some people Cloud is too slow.
Account management - Cloud only supports Cloud accounts and Google accounts. No Active Directory or other user manangement systems can be used.
Data protection - I don't know about other areas of the world, but UK and European data protection laws make it totally illegal to use Cloud for certain purposes (mostly related to the storage of personal data)
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