I should preface this by saying that most of this is fan base reasoning, since Bulba doesn't give a definite answer.
Because Bug types were too under powered. They had 4 weaknesses and 4 types that resisted it. Though Gen II onwards, it had 6 types that resisted it. With the introduction of Dark and Steel, everything was shifted around so it would make more sense. Bug is x2 against Dark because they thrive in darkness, and only x1/2 against Steel because that's Steel for you.
Bug was also super effective against Poison in Gen I, so they more or less just evened things out by giving Bugs a Poison neutrality and Poison a Bug resistance.
It also didn't make a lot of sense to make Poison and Bug supereffective against each other, since a lot of real life bugs are poisonous, and real life animals is what a lot of Pokemon are based on, notably the original 151.
If Poison was still x2 against Bug, it meant that Pokemon such as Parasect and, later on, ones like Leavanny would be x4 against Poison. The Grass/Bug typing already has x4 weaknesses against Fire and Flying, so it's underpowered as it is. Parasect and Leavanny get so little use, imagine if they still had that x4 Poison weakness. Smogon would probably have to make a tier below NU just for them.
Poison became a more defensive type rather than offensive, and since Poisoning happens to all types except Steel, Poison is actually rather overpowered. Only a few Poison type moves do not have the chance to poison their targets, which means that even without Toxic, a Poison Pokemon still has a high chance of statusing its opponent and stalling things out.