I'm not an entomologist and this how I would phrase it, in general, as a layman:
I'd call both of them "teia de aranha" (or simply "teia"), with a small difference between them. A spider web, in the singular, is more likely to be preceded by a determiner and I therefore would say "(uma/aquela/essa/esta) teia de aranha". Then again, for "spider thread/silk" I wouldn't use the preceding determiner at all and would say "teia de aranha".
e.g.
- "Olha ali, uma teia de aranha."
- "Vê só o tamanho daquela teia."
- "O local estava cheio de teias de aranha" (plural)
compare with:
- (a) "O que é isso no teu braço?" (b) "Ah, é teia de aranha."
- "Esse fio que grudou no meu rosto parece teia de aranha."
A detailist, however, would make it a point to say "um fio de teia de aranha".