I'm in complete agreement with Lars on this. So, let's start by looking at what it probabaly isn't. It is probabaly not plugs and/ore wires. Plugs and wires have an easy time firing when there is little cylinder load and generally show their problems under heavy load. So you can use a rule of thumb that unless it misses going up an on ramp at full throttle, the plugs, coil and wires are probably fine.
Oh, side comment about ignition systems first.... If the ignition system is working properly and the timing and vacuum advance is correct, you cannot get anymore horsepower out of some zoomy electronic system, so don't expect to bolt on the Wondernition and find 50 more horsepower. With that said, my preferred ignition is the Crane Electronic distributor with the dip switches that let you set 3 vac advance curves and 9 mechanical advance curves. I use it with an MSD 6AL with the rev limiter. The distributor is a snap to adjust and the rev limiter is cheap insurance.
Anyway, back to your problem. It sounds like one of a couple things may be wrong. First, you may be getting enough fuel via the accelerator pump to get it lit, but the idle and/or transition circuits are not feeding enough fuel to keep it lit. I share Lars' opinion on the AFB. They make a great doorstop, but only a very few people can make them work. So, it may be time for a new carb and get rid of the adapter plate that screws up airflow and can make fuel precipitate out. The fuel level in the float bowl may be too low or the idle mixture screws may be screwed in too far.. You should be able to see the accelerator pump work by cycling the linkage (with the engine off) and seeing two fine streams of fuel being sprayed into the front side of the carburetor. If, once it is running, you can keep it lit by tapping the gas pedal, which triggers the accelerator pump and artificially feeds it fuel,(rather than the idle circuit providing it) then you can be pretty sure it is starving for gas.. If so, highest probability is that the carb is not working on the idle circuits. There is a possibility that the fuel pump may be weak and/or the fuel filter may be clogged, but that usually shows up at high speed, not low. Oh, you do know there's fuel in the tank, right?
The next possibility is that there is a massive vacuum leak in the system. If something like the power brake hose is off, it won't idle. An engine takes very little air to idle but that air has to move at high velocity as it passes the idle discharge or you get inadequate fuel to run it. If most of the air is coming from where there is no fuel added to it, it won't run. Bear in mind that most carburetor circuits work by the venturi effect. The air passes by the discharge area and the resulting vacuum pulls the fuel out of the carb. The accelerator pump is about the only case where fuel is forced out of the carb. You have a carburetor with a gasket under it which sits on an adapter plate. Thaat adapter plate has another gasket between it and the manifold. Three things bolted together with two gaskets is not a good plan, so the adapter plate should be thrown farther than the carb.
The last possibility that I can come up with is that you've changed the timing and it is too retarded to run at idle. If that's the case, get it running and check the timing at about 1500 rpm. It should be about 15 or 20 degrees advanced at that point. So check it out at report back on what you've found , even if it is fixed, so the rest of us know what was wrong.