A Flow Action will allow a process to progress to the next step upon an action taken, hence "flow" action.
A Local Flow Action allows a UI to be displayed to the user so that an action can be taken for the case, but it does not progress the process, that's why it's "local".
For example, if I'm investigating an insurance claim case, I may want to attach pictures or documents of the investigation. I'm still performing the same task, i.e. investigating. However, I can now run a Local Flow Action (or Local Action) to display a UI to allow me to attach files. At the end of attaching my files, I am still performing the same task, i.e. investigating.