Would points to a present or future hypothetical: I would call if I had her number. Would have marks an unrealized past: I would have called, but I lost it. For past comments, you will often see both: I would say and I would have said.
Two pairs, minimal edits
If we left now, we would miss the train. A present or future picture; the speaker is still weighing whether to go.
If we had left, we would have missed the train. The chance is gone; the speaker is commenting on a past event that did not happen.
I would help you move on Saturday, but I am away. A polite, imagined offer about the future, blocked by a present state.
I would have helped you move on Saturday, but I was away. The block's time has changed, and the verb form changes with it: would have helped marks the help as a past offer that never happened, because Saturday is already gone.
I would love to join you tomorrow. In a message sent the day before, would fits naturally.
I would have loved to join you, but the date slipped past me. Both can work when commenting on something recent; the sender picks would have once they know the moment has passed.
When the pair breaks apart
A quick meaning check helps here. Ask yourself: am I imagining a past outcome differently, under different past conditions? If the answer is yes, only would have fits. I would have caught the flight if the meeting had ended earlier names a finished event you are picturing differently. With bare would, the flight would sound as if it were still catchable, which is not what you mean. One nuance: when you are simply reacting to something that just happened, both forms can be acceptable, as the article shows elsewhere. The clearest case for would have is a clearly finished past event whose outcome you are reimagining.