"But the math still doesn't figure.
$300 plumbing job.
+$200 for "free" AC & Heat service and other discounts (joining the club) = $500
Minus $100 discount on the plumbing job = $400.
So, in actuality, I paid $100 for each AC and Heat service. Along with possible discounts for anything else that I might use this company for.
Where did plumber come up with only $50 for AC and Heat service?"
The total cost for total work is easy to consider. Without the club, you pay $300 for one plumbing service. With the club, you pay $400 for one plumbing service, one heat inspection and one A/C inspection. So in the plumber's description, for $100 more you get two inspections. You're considering the heat and A/C inspections to be one item for $100 so it's just a difference in terminology.
As to whether it's a good deal, that depends on how often you use their services. Not counting the inspections, you'd have to save the cost of the membership every year in discounts, and based on what actually happened that would mean you'd have to need basic plumber's services twice in a year to break even. Keep in mind that services that cost more than the basic visit fee would make it more likely to be worth it, and since their club includes heating, A/C and electrical work, the club would save you on those services too. That said, if you don't have a serviceperson doing something on those things twice a year, it's probably not a big bargain, and since your furnace is brand new it's even less likely to be a benefit. But one other thing to consider is that (assuming services are on a sliding discount) it'll work a bit like an insurance policy, so if your A/C goes and needs major repairs you might save more than your membership fee (or even several years of membership fees) in one shot. You'll have to examine the paperwork to see if that's the case.
Only you will know for sure how often you need professional help with these things, but a quick calculation is to consider if you need service of this kind twice a year or more. If you usually do, stay signed up. If not, cancel.