ESX and ESX Server are the same thing, but I think recently VMware has decided that ESX (without the "server" moniker) is the official term. Something like that.
ESX is an installed software suite that can either be managed by VirtualCenter, by a standalone VirtualCenter Client, via a built-in web service, or via CLI (ssh). If I'm not mistaken, ESX is licensed per server, in addition to agent licenses for connecting to a VC server.
I do believe that VMware is phasing ESX out, in favour of ESXi for all dedicated-hosting scenarios.
The major difference between ESX and ESXi is that ESX has a Service Console (SC) for management which is essentially a 2GB RHEL application. ESXi on the other hand does not have a service console and is therefore much smaller (~32MB) and more secure. ESXi is the format of the Hypervisor/Kernel going forward. ESX will be available for about another year to allow partners and customers to transition any applications running on the SC to CIM or some other app framework. The New name for the Hypervisor starting with ESX/ESXi 4.x will be vSphere. Hope this helps.
ESXi management is GUI only. ESX management can be GUI or CLI based.