It really depends on what you're growing, how you're growing it, what you expect from your garden, how big the weeds are, how close they are to your plant, how many weeds there are, how fertile your soil is, how the specific weeds interact with your plants and each other, etc.
Plants can compete with each other, produce chemicals to inhibit other plants, and stuff like that (but they might do beneficial stuff, too, in some cases). Calendula can kill nitrogen-fixing microbes that are important for beans. Black walnut can inhibit lots of plants; same for wormwood. Sunflowers are said to inhibit plants somewhat.
Muskmelons will smother most weeds, in my experience (so if you're growing those, I wouldn't worry a terrible lot). Watermelons let more weeds through (and grasses, if present, can be problematic); watermelons still grow pretty well with weeds, though (but if you like to pull weeds up, grass is hard to deal with there, as pulling it up can disturb the watermelon roots).
Here are reasons to weed:
- To prevent more weed seeds from spreading (a single weed dropping seeds can mean loads of them next year)
- To prevent weeds from choking out the plants (not just competing for nutrients)
- To be respectful to your neighbors (neighbors don't usually want your weeds spreading to their yard)
- It might be illegal in some areas to let weeds grow.
- It's harder to navigate and find stuff in a garden full of weeds.
- Weeds sometimes get diseases (which in theory could spread).
- If you container-garden, weeds are a bigger problem (if you have weed seeds in your containers, which not everyone does), because nutrients in containers are in more limited supply, and so is space for roots.
Reasons not to weed:
- It's extra work. Is the extra production worth it to you?
- You can learn how to grow vegetables better with weeds. If you want to know if it works, asking people who always weed their gardens isn't going to give you the answer. You need to find someone who has really tried to grow vegetables with weeds, and learned all the tricks to assist in doing so (or you need to teach yourself).
- It's good for the insects and biodiversity of your garden. Weeds can be great food for grasshoppers, or whatever, if you like to nurture them. Butterflies will lay eggs on weeds. I saw a whole bunch of wasps enjoying purslane flowers, today. I was wondering what was attracting so many of them to our garden (I think it's the purslane, as it has become a frequent weed, especially in our containers).
If you don't want to worry about weeds very much, growing in the ground with black plastic over it keeps most of them away. You still have to weed right around the plant, though (if you weed), but it's a lot less to worry about. Shredded wood mulch can also keep weeds out, but it's more expensive and needs to be put down more often; however, mulch can keep weeds out from around the base of the plant, too, whereas black plastic can't. Mulch nourishes the soil, and you can water all over it instead of just where the plant is.
I personally believe that you can breed and acclimate plants to grow better with weeds, but I don't know anyone who is pursuing this, currently.
You can get a good harvest with weeds--but weeds can prevent a harvest altogether, too. It depends.