What does leveling your equipment get you

There are two sides to this, depending on whether you plan your path or not:

Because the Clockworks is constantly moving, even when you're progressing through a level, you have the option of deliberately waiting to step onto the elevator in order to reach desired levels. Each of the levels has a (usually) pretty obviously descriptive name reflecting the types of monsters you will encounter, and since gear with fewer resistances is stronger against each source, you can "stack" your resistance to a select few and plan out your path for maximum resistance. They hide the option to show the gate map under the upper-left menu with no way to hotkey it, but it will show you where the elevator is currently pointing.

For the more casual player who expects to face damage from a variety of sources, pieces with multiple resistances will help negate more damage on average than a specific piece. Because they have more individual resistances, there is a greater probability of entering a level they can resist some of the damage from, rather than limiting all of your potential resistance to a single source and missing it.

The bottom line is that very few sections of the game actually depend on your gear's type enough that it makes a significant difference. As long as you have enough stars on your gear and a fair amount of experience with the enemies' tactics, you can just pay attention to your "threat meter" to the lower-right of the minimap.

Chances are, they're referring to Lord Vanaduke, who is a boss down in the Firestorm Citadel down in tier 3.

The party member who bailed probably didn't want to spend his energy without making to a boss.

The others were probably lamenting that they had to work their way to vana instead of being able to catch a ride with a party that was almost there already.