I have an April Rain Bulbori growing in one of my pots, but it keeps wilting and getting parched. All my small baskets of enrichments give me Weed-B-Gone. Do I actually need to keep it in good condition? If so, how does it affect the plant?

plants do grow slowly on their own (as in real life). Other people can attend to your plants, which is why they often get posted in region chat- if you don't have the right item, other people might.
By tending to the plant, you speed up the growth, and also edge towards bonuses for the pet- you'll notice the bit that says "Bonus: 0/2 (46%)"
Bonus 1/2 is extra stats, bonus 2/2 is an extra carried trait (for non-majestics, anyway. for Majestics they get a second stat boost)

Compare the stats on a regular unboosted exclusive pet to a pet with the full 4/4 root strengthener boost + the Bonus boost (I believe the root strengthener boosts are about +5 each, and the bonus boost is about +10)

Thanks for the advice! Btw, I harvested it today!