First off, to be clear, LENR or CNMS or whatever other name you give it, cold fusion is still entirely theoretical, and probably still not possible. It's not nearly as flawed as the "cold fusion" work from decades ago, but still considered fringe science.
For Independence Day's sake, lets assume that some theory of LENR is valid and we can make a weapon out of it. For starters, the "cold" or "low-energy" term in the name doesn't reflect the energy output of the reaction, but the energy input. Naturally occuring "hot" fusion takes place inside of the cores of stars, because it requires incredibly high temperatures and pressures to overcome the natural repulsive forces that keep nuclei apart. In a theoretical cold fusion model, some mechanism allows us to trigger fusion to happen at room temperature.
To see why you might want to use such a device, you'd have to look at what the alternatives are: conventional explosives, fission weapons, thermonuclear weapons, or "pure hot" fusion weapons.
- Conventional weapons are what "normal" bombs are made of, based on chemical reactions; these are highly inefficient compared to nuclear reactions, so it would take a lot of mass to make a powerful explosion.
- Fission weapons are what real-world nuclear weapons are: uranium/plutonium undergoes uncontrolled nuclear fission, releasing a lot of energy quickly. These are orders of magnitude more efficient than chemical-based explosives, but fission reactions are less energy-efficient than fusion ones. Plus, the materials to make them are expensive and deadly to handle, and they leave radioactive fallout behind.
- Thermonuclear weapons use a fission bomb to produce high enough energies to trigger a second stage fusion bomb. Hydrogen fusion is the second most energetic nuclear reaction we know of (second only to antimatter annihilation), so these are very powerful. But because they require a powerful fission bomb to work, they suffer from the same expense, health risks, and radioactive fallout problems as fission weapons.
- "Pure hot" fusion weapons don't yet exist; in theory they would work on the same principle as the sun: induce hydrogen fusion by way of extreme temperatures and pressures. These weapons, if possible, would by very powerful with almost no dangerous by-products, as the bulk of hydrogen fusion's output is just helium. Hydrogen is also abundant in the universe. The problem is triggering the reaction: outside of wildly expensive lab setups, the only thing we have that can trigger fusion on cue is a fission bomb, as above.
If the Independence Day world has figured out cold fusion, it would be a very good choice for a weapon of this nature. It would have all of the benefits of a "hot" fusion bomb, without the problem of triggering it. Since we don't know the mechanism it takes to produce such a bomb, we don't know how expensive it is, but the material to make it was obviously available to the US government in-universe. Once the bomb's been manufactured, though, using it would be much easier, since it would operate at normal atmospheric conditions.