Generally, tyres make more difference to how the bike feels than wheels. So did you use the same tyres at the same pressure on each? Then are the rims the same width, as a different width will make the same tyre feel a bit different?
But if the bad wheels are bad enough, they can make a difference - what others said: poor spoke tension, out of shape in both dimensions, cracked rim, worn out hubs, etc.
What is 'poor climbing' - just slow, or something else as well, eg how it feels?
Heavy will clearly be a bit slower to climb, but you would need to measure it quite carefully to detect it - weight doesn't make that much difference if you are not racing. People often talk about rotational weight being massively important but, in absolute terms, the forces are tiny (put the bike in a stand and see how hard it is to make the wheel rotate with one finger...)
All things being equal, I can't immediately think what would make the 'bad' wheels feel worse on bumps though - unless it was different tyres or pressure. Poor spoke tension would give some suspension which should actually help with that. Maybe egg-shape rim deformation could add in some extra vibration? Possibly a loose hub or trashed bearing could feel worse on bumps...?