Both are tapered pipe threads, with the same taper (1/16" in an inch) but 1/8" NPT is 27 tpi, with a major diameter of 0.405", whilst 1/8" BSPT is 28 tpi, with a major diameter of 0.383" - note those diameters carefully, the NPT fitting is a hair larger than the BSPT one.
They are almost indistinguisable to the naked eye, but if you try to screw a 1/8" NPT fitting into a hole meant for 1/8" BSPT you'll typically get it about 1/2~1 turn in before it jams, on the other hand, you can screw a 1/8" BSPT fitting into a 1/8" NPT hole without ever realising they are not meant for one another.
I've heard of folks "re-tapping" the 1/8" BSPT hole to take the 1/8" NPT fitting, and also of them just forcing the fitting and letting them cut their own threads (more likey deforming the fitting slightly).
What I do is buy 1/8 BSPT to 4mm compression fittings on the UK ebay site, and swap the 4mm ferrule (or olive) for a 1/8" one.