The connection is a heroic sized taper compression fitting. The manifold is the male part and the sealing surface describes the side of a 45 degree cone - and the pipe is the female part which is held in place by the two piece coupler.
Getting the two tapered cones to mate precisely is the problem. Problems arise with the manifold portion in that the surface can be corroded, or the manifold may have "dropped" because of heat/age/gravity/loading by the muffler pipe and caused a mis-alignment.
Or the pipe might be a later replacement where the installer did not bend the pipe properly to produce proper sealing. This is one of those parts you frequently have to "tweak" the alignment by strong arm methods, or by hammering or swaging the female cone portion of pipe.
The retailers sell a "gasket" materials for use in this area - but my experience is that gasket WILL NOT stay put and you end up chasing yet another leak and doing the job all over again. Much better to get the proper metal to metal seal, and use a little pipe dope to lubricate the fit, and grease the clamp application. Heating will cure the pipe dope.
Also reproduction clamps are known for being "undersized" and don't fully hold the pipe to the manifold. A good original is probably your best bet, if one can be found.
Hope this helps.
Joe K