Drip is more concentrated in one respect, given the same volume of water and measurement of grounds: a French press keeps the oils whereas drip filters them out. But that also means you lose out on some of the richness (and potentially cholesterol-raising compound cafestol, among other compounds) when opting for filtered coffee.
In terms of flavor, generally the higher temperature, longer steep time, and higher pressure methods extract more out of the beans. It's balancing act, though, because you'll end up with an unpalatable brew if you try to do all three, or use too little or too much solute (grounds) or solvent (water). You don't want to over or under extract the beans; not everything in coffee is pleasant to the taste buds (it is roasted beans, after all), including caffeine which is naturally bitter.
The bottom line is that you'll get more "coffee" in your coffee if you go with a French press, or even more with an espresso maker/moka pot, and it will be by far better than drip if you nail the other factors.