vSphere is a general name for the technology (also used for the name of the Client console) , vCenter is a specific server based application used to manage multiple ESX hosts (which you still connect to with the vSphere Client)
As such your question should really be about the differences between "via vCenter" and "direct from ESX host" backups - and the most obvious reasons are
a) Must use vCenter if an HA (High Availability/failover) setup is configured in your ESX environment (if you go direct to host the backup will fail if the virtual machine has been failed over to a different host)
b) Can be useful to use vCenter for ease of job and selection management
c) You can sometimes see speed differences between a direct host backup and a vCenter backup (may be down to the environment as to which is quicker so you would have to test)
d) Ability to integrate the security of the backup login accounts via Windows credentials if using Windows vCenter server
e) However if you have a vCenter why not use it, if you do not have a vCenter then you have to use the direct from host option
