The difference between the two is the size, generally with the barrique being your "standard" 225 L (60 gallon) barrel, and the puncheon being 500 L (roughly 133 gallons) in size. That's the difference in the products, but the effect each has on wine, over time, is different.
Assuming they are made from the exact same oak, same thickness, etc, the barrique will impart goodies into the wine more quickly than the puncheon. The effects of micro-oxygenation are also realized more quickly in the barrique vs the puncheon, all for the same reason. The reason is that in smaller vs larger vessels, the volume of wine to surface area of wood contact is greater in the larger vessels than the smaller ones. Smaller vessels with identical wood profiles will always "do their thing" much more quickly due to this factor. Problem is, if you hit the flavor threshold too quickly, like in a new 6 gallon barrel, you may not have the wine in the barrel long enough to take advantage of the effects of micro-oxygenation and concentration.
For instance, we might only leave a wine in a new 6 gallon barrel for just a few short weeks before the oak flavor level dictates that it's time to remove the wine, whereas you might leave wine in a 60 gallon barrel for 1 to 2 years before the oak flavor gets to the point where you're ready to pull it. Leaving wine in a 6 gallon barrel for 1 - 2 years will over-oak the wine, and probably cause it to become oxidized as well. Leaving wine in a 60 gallon barrel for a month will probably have negligible effects upon it's taste, and little micro-oxygenation will be realized.
The time you leave your wine in a barrel must be right-sized to the barrel volume itself in order to achieve the desired results. We pretty much already know from experience that a year or two in a 60 will provide excellent oaking / microx / concentration opportunities for a wine. With the 133 being so much larger, it may take several more years to hit the oak level of a a 60, if it ever does, and you'd also need to worry about too much oxygenation, thought this is only my speculation, having zero experience with puncheon sized wood barrels.........................