What is the difference between ILS and VOR/DME?
Which one is better for carrying out approaches?
An ILS approach is more precise for two reasons:

The ILS localizer is more sensitive than a VOR radial, therefore providing more accurate lateral guidance.
The ILS has a glide slope providing vertical guidance. A VOR approach does not have any vertical guidance (although with a VOR/DME one can at least determine the exact point at which to start a CDFA (continuous descent final approach) and regularly crosscheck the altitude with the DME).

Why do pilots use VOR/DME when the ILS is a better option?
Pilots fly a VOR/DME approach when they are cleared for the VOR/DME approach by ATC. Why would ATC provide a VOR/DME approach clearance when the runway also has ILS? This is usually due to ongoing maintenance of one of the ILS antennas. When an antenna is not available, you obviously cannot fly the ILS approach (although it may be possible to fly a localizer approach, if only the glideslope antenna is unavailable).
A pilot can also request the VOR/DME approach for various reasons, e.g.:

It could also be that the aircraft is not equipped with an ILS receiver antenna and is therefore unable to fly an ILS approach, but at least for an airliner that would be quite unusual.

For training the specific approach type or to maintain proficiency (thank to StephenS for pointing this out).

Sometimes flying the VOR approach might actually be more convenient if the VOR approach has different approach transitions available compared to the ILS. In Jersey (EGJJ) for example, the VOR is not co-located with the aerodrome and therefore allows an IAF further to the east compared to the ILS approach, which begins at the NDB (which is co-located):