They’re interchangeable S/N for conditional (‘would (be)’) - but byddwn also has a non-conditional meaning (habitual imperfect) which is countrywide. In other words, even the gogs use byddwn when they mean hab impf.
Conditional:
N - (Ba)swn i’n mynd yno tasai gen i ddigon o bres
S - Byddwn i’n mynd yno pe bai digon o arian 'da fi
I would (conditional) go there if I had enough money
Hab impf:
N and S - Byddwn i’n mynd bob bore i’r dre
I would (=used to) go into town every morning
Thank you for clarifying re habitual past (something I was explaining in an ESOL context just last week 
) but I was sure I’d come across something else… And I’ve found it! – L2 Welsh 24n, from start: e.g. Byddet ti’n dweud wrtha’ fi, fysat ti? has byddwn and baswn forms in the same sentence, both translated as ‘would’, both looking more conditional than habitual to me. Is that just elegant variation, or what? Or should I be paging @aran and @CatrinLliarJones to account for themselves?
Yes, as I recall, more conditional in the challenge. Going back to English, I think it’s possibly also a Northern thing. I like it because it feels less forceful. I’ve been picked up for it in work (Bristol area), for saying “I wouldn’t say that” instead of “No, you’re wrong” or similar.
Combination there of a very deliberate decision to mix stuff up à la stryd, triggered by a ‘staying within the boundaries of what we’ve already introduced’ situation…
Yes.
Love the way that there’s pretty much always an answer to any SSiW-type question that pops into my mind, somewhere on this forum.
Today it was the turn for Baswn/Byddwn i! 
Thanks for the above.
Here’s a good example
I always get these mixed up and confused between baswn i and byddwn i.
Excuse my grammar (and please correct it if it’s wrong),
Taswn i yn dy le di, baswn i’n gosod hynny yn ôl.
I’m trying to say, ‘If i were you (in your place), I would put that back.’
Yes, that’s right James.
Just as a tip, you might hear people shorten it to 'swn i yn dy le di, 'swn i’n gosod hynny yn ôl., but it’s exactly the same sentence.
And if you heard someone saying the same thing with the ‘byddwn’ form, you’d get pe byddwn i yn dy le di, byddwn i’n godod hynny yn nôl
Diolch yn fawr, Siaron.
Roeddwn i’n ansicr about the South equivalent of byddwn, but now you’ve told me without prompt!
Pe byddwn i = Taswn i
Wi’n meddwl 
Yup, that’s it!
taswn is actually pe taswn, but you rarely hear people using the pe in speech - it’s quite a formal form now.
Diolch,
So regarding your southern example
“pe byddwn i yn dy le di, byddwn i’n gosod hynny yn ôl”
What change is there, or mutation, to make the Southern version an ‘if’ version, as there is PE before byddwn.
You say that Pe is formal and not really used in everyday speech.
Mae’n flin 'da fi am ofyn llawer o gwestiynau. 
The pe is used in the Southern form (pe = conditional ‘if’) because there is no mutation so “if I would” needs to be distinguished from “I would”.
The Northern form comes from what was originally pe+yd+byddwn. In the South just the yd got left behind but in the North, byddwn became baswn somehow (so you’d have pe+yd+baswn) and what’s happened over time is that the d in yd became a t (kind of an unofficial reverse mutation!), so then it became petaswn, and then shortened again to just taswn.
Actually, in the south, it’s not unommon to mix the forms, e.g. in the southern course you’ll hear examples like:
byddet ti’n dweud wrtha i 'set ti’n gwybod yr ateb
bydden i’n dweud wrthot ti 'sen i’n gallu