These two seem to be an intermediary for external network and internal network. What is the correct use for each deployment? What are their differences?
"Jumpbox" (along with similar terms such as "jump server", "hop box" and the like) is meant to describe a server that sits between a client or management system and the target server or system being managed, which a user or application logs into from the client, and from there initiates a second session to the target server. There is no implied directionality with respect to whether the client and server each exist within a given network perimeter or outside of it, but generally they will be on opposite sides of such a perimeter and thus unable to directly reach one another using the desired protocol. Typically a command-line-based protocol such as ssh or telnet is used.
"Reverse proxy", on the other hand, is a bit more specific. A "proxy" server is defined with respect to a specific protocol or set of protocols, typically web (HTTP/HTTPS) and/or FTP, and allows the client (browser) within the network perimeter to access resources outside the perimeter by making connection to the proxy and asking the proxy to access the external resource on its behalf. A "reverse proxy", therefore, operates in the opposite direction: the client is outside of the network perimeter and the resources are within.
Many TCP-based network protocols can be relayed in this manner, and it's very much a matter of arbitrary convention which term is used for which set of protocols.