First, not to chastise, but rather inform, there is no need to remove the distributor to install new points. There are 2 notches on the crankshaft pulley, and a pointer on the timing gear cover. Since it now seems to be out of time, bring #1 cylinder up on TDC on compression stroke. Pull #1 plug, with a finger, or your thumb, bump the starter until you feel compression on the #1 (front) cylinder. Watch for the first notch on the pulley to align with the pointer. You should be able to get it dead nut on, by tightening the fan belt, and turning it by the fan blade, to get it aligned with that first notch.
Then check the position of the rotor button. It should be at the approx. 2 o'clock position. If not, loosen clamp bolt, and rotate to that approx. position, and the highest point on the distributor cam lobe. Set points at .020.
With transmission in neutral, and brakes locked, and #1 plug grounded out on the block. Turn ignition switch on, and rotate distributor until you hear/see a spark, then turn off ignition switch, and snug the distributor clamp bolt down. You should be dead nut in time. If everything is in operating order, it should start right up. You can always check the timing, with a timing light, if you happen to have one, just to double check the timing.
My first thought was you may have the cap 180º out, but you'll normally get a backfire when doing that, firing on the exhaust stroke. Most caps of quality have a #1 encircled on the cap, marking #1 cylinder plug wire hole. If not, #1 plug wire hole is approx. 2 o'clock, when facing it. Note rotation of rotor button when the cap is off, and make sure the plug wires are in proper firing order of 1-3-4-2, which is stamped in the engine block, near the cap.
What I refer to as lugs, are the pick-ups inside the distributor cap. If the cap has been on there for a good many years, you definitely want to look at those lugs. They can get a carbon like buildup on them from ionization, which can through you out of time a few degrees, but can easily be scraped off with a pen knife and you'll be good to go, or, simply replace it. You'll want to lightly scrape, or buff the rotor button with fine sandpaper, to a bright finish. It doesn't take much. I've been running the same caps on my one Super C, and 240 Utility for 20+ years. That buildup seems to be worse in years where we have higher humidity.
If this doesn't solve your problem, let us know...
** Edit** By the way, when installing the screws in the points, one of the handiest tools to do that is one of those short screw starters. You'll want the short one, so as to clear the oil filter. And spring for the one with the magnet on the opposite end. I keep one in my tune-up tool box, just for that purpose.