In its original meaning, it can be exactly the same. You also have the gender neutral "løsunge" and "uekte barn", or just straight-up "barn født utenfor ekteskap".
The first two are rather derogatory (the degree depends on dialect), the third is passable, and the last one is your safest bet if you don't want to offend anyone.
Either way, having children outside of marriage is hardly a big deal in modern Norway, so it's not something that comes up a lot in conversation.
It can be used negatively, but doesn't necessarily have negative connotations.
Its use in modern Norwegian is very limited, so I wouldn't use it as a general replacement of "small" or "weak". Aside from the compound "spedbarn", you'll come across it when people talk about "spede begynnelser" or "sin spede barndom", referring to early beginnings and one's early childhood respectively - with the implication that things are often smaller in the beginning.