Well, without getting into grammer it is the same difference as in 'at' and 'beside':
"The knife is at the plate - Kniven är vid tallriken"
"The knife is beside the plate - Kniven är bredvid tallriken"
'Bredvid' is a more specifc word than 'vid' and can only be used to pinpoint a location (or similar) whilst 'vid' can be used several other ways, not only locationwise....the same way you can use 'at' in english.
On a geographical scale, "vid" can be translated as "on", as in "vid kusten" (on the coast) or "vid gränsen" (on the border). Its meaning, though, is still "near" or "next to". So, in Swedish this usage is consistent with "vid bordet" (at the table) or "vid dörren" (at the door). It is a peculiarity rather in English that we switch between "at" versus "on", depending on the context.
In regards to translating "Kniven är vid tallriken", although "The knife is at the plate" may be a little unusual, it has the correct meaning, for which we would more likely say "The knife is by the plate". "On the plate", on the other hand, would be "på tallriken".
I saw an explanation in one of the discussion threads for this lesson about how the size and position of a person or object relative to another can often be a helpful way to differentiate between "vid" and "bredvid". Could a moderator / native speaker kindly comment on this?
Are you more likely to use one of these prepositions with two people or objects of similar size, and the other preposition when reporting on the position of a smaller object relative to a larger, perhaps immobile reference point?
How interchangeable are they? I'm sure there are some more or less fixed idiomatic expressions or grammatical constructions where you must use one or the other. Tack så mycket in advance for your help!