Compering between ILS / BARO-VNAV
ILS provides both lateral and vertical guidance with high precision, which means aircraft using an ILS are allowed to get very close to the ground and each other without visual contact; in theory, a Category IIIC ILS is precise enough to land with zero visibility, but so far no examples actually exist.  Where ILS approaches exist, they're usually preferred.

Baro-VNAV provides only vertical guidance; it would be used for RNAV procedures in conjunction with some form of lateral guidance such as DME/DME/IRU.  Due to the pilot having to manually enter the altimeter setting and the general accuracy of altimeters, the margin of error is much higher and aircraft are not allowed to get nearly as close to the ground (or each other) as they would be with ILS.  Also, Baro-VNAV systems are known to be inaccurate at extreme temperatures (either high or low), so they aren't always available, whereas other vertical guidance systems are immune to that.

The other main difference is cost.  ILS requires both localizer and glideslope equipment on the ground for each runway served; installing and maintaining those systems is quite expensive, whereas the on-board equipment is relatively inexpensive.  In contrast, Baro-VNAV is already installed on many airliners and requires nothing on the ground, just minor modifications to RNAV procedures already being developed for GNSS-equipped aircraft.  The accompanying lateral guidance systems also already exist for other reasons, so there's little or no cost there either.  The combination allows airlines to more reliably and more safely serve the public at more airports with minimal cost to everyone, so it's a reasonable compromise for airports (or individual runways) that don't have enough traffic to justify the high costs of ILS.