I don't have enough rep to comment but I'm going to expand on what RuBiCK said.
In the first case, if you have a password for JIRA that I'm going to use to log in to the website with and fiddle with some tasks, then this goes in LastPass. You can store it there and use the extension or the vault to help me remember a complex password. You can also use LastPass to send credentials to my team members and then they can use LastPass to log in with.
In the second case, if you have made a script which asks AWS for what instances you have running right now then that script is going to need some credentials to log in with. You could put them in LastPass if you wanted but every time you went to run the script it would mean you have to go and get them from LastPass, enter them into the script somehow and get my list of instances.
Vault however presents an API which your script can use to ask for credentials when it runs. The credentials are stored in a secure way by Vault and you can now store much more in there as well.
The above is also why Vault wouldn't be useful for most people to do what they want to do. You don't want to have to write something to talk to Vault and create browser extensions etc when LastPass is, kind of, affordable and good at what it does.
I hope this helps!