Vagrant is more resource intensive since it spins up a virtual machine running another OS with dedicated filesystem and memory. It is slow, but was the only game in town before the faster-running Docker came along.
Docker "usually" runs on the same OS, but shares memory (though isolated), so the processes it contains should not interfere with anything outside the container. I say "usually" because running Docker in Windows or MacOS actually does use a virtual machine to host the Docker images.