The people who call themselves oddfellows are members of various friendly societies, but not all friendly societies call themselves oddfellows.
Friendly societies, by the way, were all that working people had to fall back on in the days before the NHS and the welfare state. They evolved out of the old medieval guilds and remain free associations of folk who band together to pay contributions to a fund that will cover them in times of sickness, unemployment and when they get to old age.
The benefits are modest by modern standards. The nation's largest group of Oddfellows is the Manchester Unity of Oddfellows, with 120,000 members and 151 branches. This giant among friendlies was founded in 1810 (it celebrates its 200th anniversary tomorrow) and can trace its origins back to the exile of the Israelites from Babylon in 587BC, so they say.
Nowadays the "social" side of the society is more important than the benefits. No one knows where the name "oddfellow" comes from.