What's the difference between 开 and 举办 when used for the meaning "to hold an event"?
开 has other meanings beside; 'to hold' (events)
举办 is a specific term for 'hosting' and 'organizing'. If an organization 举办 something, you expect it to be well organized.
开 is less formal than举办. E.g. 在我家开派对 is less formal than 在我家举办派对.
在我家开派对 can mean you casually call a few friends over to your house. Not much preparation is required

We say 开班会, not 举办班会 because 班会 is a regular event, no additional organizing is needed

We say 举办学生会选举, not 开学生会选举 because 学生会选举 is a formal event that needed to be organized


In short, 开 (to hold) is more general and casual; 举办 (to hold) is more specific and formal (officially hold and organize)
开 itself carries the meaning of 'hold', it doesn't need to be short for anything
Depend on the context, 开 can be short for:

开办 (open and operate/ open and run) e.g. 开(办)学校

召开 (call for and open) e.g. (召)开会议