In short, price (Enterprise is a subscription you pay per user) and features available.

Find a comparison here: https://www.odoo.com/page/editions

For most businesses, the key difference is in the finance and sales capabilities of both editions so you may want to pay special attention on the details there.

If you want to see the differences in action, you can hop into a (shared) runbot-instance of both, Odoo Community and Odoo Enterprise: https://runbot.odoo.com

It is possible to start with a Community instance (self-hosted / hosted by a service provider of your choice) and upgrade to Enterprise later*. The other way is, while in theory technically possible (given enough time), a struggle you don't want to waste money on.

An Enterprise database can be hosted by Odoo directly.

*) Whether or not this is meaningful depends on your use cases and whether or not customization is required. This is especially true for customization needed for Community setups only.