Interesting topic. Perhaps it was about time we asked other people too.
Well I haven't ever had trouble with the weeds, and didn't even notice if a plant wouldn't grow when it had weeds on them, but I know that weeds do
not affect the harvest amount. Every time I send my sim to harvest his money trees (and I only do it when they are all Harvestable), he gets 150-185 money bags in his inventory by the time he's done. I did it again when they all had weeds on them, and still got the same amount, and sold for the same amount too.
I'm not sure about the quality, because Money Trees have a particular quality system, but it didn't seem to affect my Plasma plants or fruits at all (growing a few of them, none are perfect yet, but they are growing just normally).
::EDIT::
I checked the wiki to see if I could find anything weed-related, but there was only the same old Master Planter skill challenge, which doesn't change much. Also, does anyone know what's a Garden Gnome?
Master Farmer
Master Farmers have harvested at least 650 fruits and vegetables. The plants of Master Farmers remain watered and fertilized longer, meaning their gardens are more efficient. And if you have a garden gnome he will come to life and water your garden.