OK, but I'd have to say that there are so many variables to consider that anything I suggest of may be of limited value. However, one could take that as a fairly universal caution, so I'll try.
Heavy flows or highly variable stream conditions: I think the advantage often goes to weight on the leader because adjustment is easier and the choice of fly is greater. With weight only in the fly, one has to either rely mostly on line management or have flies tied with different amounts of weight. Relying strictly on line management can severely limit "bottom time" in some heavy drifts, and having flies with different weights only works for some sizes/types of flies (and multiplies the number of flies needed).
Light flows or consistent stream conditions: Weight in the fly can sometimes work better, provided that the weight is well-suited to the predominant flow. This still depends somewhat on the size and type of the fly being used, but line management adjustments are usually easier under these conditions.
High-sticking or straight-lining: Because both of these techniques often rely on feeling the weight ticking the bottom in order to gauge the drift, weight on the leader (or on a dropper) usually works better. Large "anchor" flies can sometimes provide that contact and feel, either alone or in tandem. However, the trade-off (especially when smaller flies are the main enticement) is often flies lost compared to additional bites on the anchor flies.
Indicator nymphing: This often depends on the type of indicator being used/required. Other things being relatively equal, indication of a strike may be improved by using just a weighted fly. There can be something of a trade-off, though, as trout may not hang onto a heavily weighted fly as long. When a dry fly is used as an indicator in fairly shallow water, the appropriately weighted fly usually has an edge. Without weight on the leader, contact is more direct and the dropper can be shorter. This certainly makes casting easier.
Multiple fly rigs: If you are going to attach a number of nymphs to a leader, then it makes a certain amount of sense to use an anchor fly rather than attaching additional weight as well. However, adjusting this rig to different flows and drifts might also require changing the anchor fly, so that can become a trade-off.
Pressure considerations: These can argue either way depending on the flies and presentation methods that the fish see most. Fish can become "shot shy" (any experienced Great Lakes steelheader can attest to this), but overexposure to beadheads (especially flashy beadheads) can also cause rejection at times. Weighted flies drift differently than unweighted flies, and that difference can be played in either direction in order to overcome resistance.
As a proviso to this, I'd add that something about the popular shotgun approach of fishing multiple "searching" nymphs in a fairly random way doesn't often appeal to me. I can't deny the general effectiveness of this approach, but it seems a bit like hedging a reluctant bet. I much prefer to fish a single fly whenever I can, and I usually base my fly selection on my best guess about feeding activity and available foods. But that's just me. I'm a hatch-matcher at heart, even subsurface.