There can be differences between ice packs, but in practice the differences seem pretty small, and you'll have a hard time finding hard data on different ice packs' cooling abilities. How much ice/ice packs cool is determined by three things:
First, the specific heat capacity of a material tells you how much energy it takes to raise 1g of the material by 1 degree C. Water has an unusually high specific heat capacity, so it has a lot of "thermal inertia", meaning that cold water can absorb a lot of energy from warmer objects before becoming warm itself. Secondly, the latent heat of a material tells you how much energy it takes to melt 1g of frozen material. Water also has a very high latent heat capacity, so ice at 0 degrees C will absorb a lot of heat before it fully melts and becomes water at 0 degrees C. Finally, since the first two quantities are expressed with per mass units, we can conclude that a more massive object is a better heat sink than a smaller one - two ice packs can absorb twice the energy of one ice pack.
In terms of common substances, water is pretty much as good as it gets for a coolant - it has a very high specific heat capacity in both its solid and liquid forms, and the amount of latent heat absorbed while melting is also very large. A block of iron, on the other hand, absorbs slightly less energy per volume than water (it has 1/10 the heat capacity, but 8x density), and since it undergoes no phase change, it absorbs no latent heat. In total, the cooling capacity of a frozen block of iron will be roughly 20% of the same volume of ice over this temperature range.
The vast majority of the cooling effect of ice comes from melting - it takes only 2.1J to raise 1g of ice from -1C to 0C, but it takes 334J to turn 1g of 0C ice into 1g of 0C water. Keep the cooler packed with as much ice as you can, and replace it once it's melted, and you should have little trouble keeping food cold with a decent cooler. The cold liquid water is better than nothing, but most of its cooling ability was spent when it melted. Ice packs will have pretty much the same result, but may be more convenient since they're reusable and won't soak everything in the cooler, although it won't be quite as obvious when they're fully melted.