No, a rump kernel is not an unikernel, check the FAQ linked from
http://rumpkernel.org/However, you can use rump kernels as a major component of a unikernel implementation. A rump kernel provides environment-agnostic drivers, meaning you can integrate them pretty much anywhere.
Now, what is a unikernel? From my perspective it's essentially:
1) application
2) config/orchestration
3) drivers
4) nibbly "OS" bits
So from the bottom, the nibbly bits include things such as bootstrap, interrupts, thread scheduler, etc. It's quite straightforward code, and a lot simpler that the counterpart you'd find e.g. in Linux. But you can't do much anything useful with the OS when that part is written.
Drivers are difficult because you need so many of them for the OS to be able to do much anything useful, and some drivers require incredible amounts of effort to make them real-world bug compatible. Just consider a TCP/IP stack -- you can write one from scratch in a weekend, but the result won't work on the internet for years. Then you may need to pile on a firewall, IPv6, IPsec, .... A rump kernel will provide componentized drivers for free. The policy of if you use those drivers in a unikernel or microkernel or whateverkernel is up to you, but I guess here we can assume unikernels.
The config/orchestration bits are actually quite an interesting topic currently, IMHO, at lot of opportunities to make great discoveries. Also, a lot of opportunities to use the rope in the wrong way.
The applications depend on what sort of interfaces your unikernel offers. If it offers a POSIX'y interface, you can run existing applications, otherwise you need to develop them for the unikernel.
Now putting rump kernels and unikernels together: the nibbly bits are straightforward, the drivers come for free via rump kernels, and those drivers provide POSIX syscall handlers, so POSIX'y applications just work. That leaves the config/orchestration stuff on the table. There's a rumpkernel-based unikernel called Rumprun available from repo.rumpkernel.org. It's essentially about solving the config/orchestration problems. Due to the rump kernel route, the other problems were already solved in a way which can be considered "good enough" for our purposes.
Hope that clarified the difference between rump kernels and unikernels.
(edit: minor formatting fix)