They are somewhat related, but it is possible to have a high WHIP and low ERA and vice versa. WHIP is a better rough estimate of a pitchers performance, while ERA shows more or less how "lucky" a pitcher has been. A low WHIP but high ERA shows that a pitcher has been very effective (allowing few baserunners by his own actions) but due to circumstances outside his control (bad defense mainly) has allowed runs that are technically earned, but aren't expicitly his fault. In this case if he keeps his WHIP the same the ERA will slowly come down to reflect the lower WHIP just by the law of averages. Conversely if he has a high WHIP but low ERA that shows that he has been putting runners on, but his defense, poor baserunning, etc has bailed him out. In this case if the WHIP stays the same he won't be getting the good breaks all season (law of averages again) and his ERA will begin to rise to reflect that.
So I guess I'm basically repeating CPH's comments in that that WHIP "causes" ERA but ERA really has no bearing whatsoever on WHIP itself. Consequently, WHIP is the only standard fantasy cat that truly reflects a pitchers effectiveness if you ask me.
And king, I see what you're saying, but the differences can be and often are very subtle.