Dawn and Fairy are, by a wide margin, the two dish soaps most recommended by the international bubble-making community, and that's no accident: both are concentrated detergents with a high load of anionic surfactant, the ingredient that actually lowers the surface tension of water and lets a film form. Dawn (a U.S. brand, Procter & Gamble) combines sodium lauryl sulfate and sodium laureth sulfate; Fairy (same parent company, but formulated for Europe) uses a blend of alkyl polyglucoside and sodium lauryl sulfate. In practice, both work through the same mechanism.
The real difference is concentration, not brand. Dawn Pro can be more than twice as strong as a "regular" or diluted Dawn, and within the Fairy lineup the anionic surfactant concentration varies by country: 15-30% in some markets, 5-15% in others. That means a bottle of Fairy bought in the UK can perform differently from one bought in another European country, the same way a Dawn "Ultra" isn't the same as basic Dawn. The dilution of the bottle in your hand matters more than the logo on it.
Which one should you pick?
If you live in the U.S. or Latin America, Dawn (ideally a concentrated version) is the option with the longest track record and the easiest to find; it's the reference detergent in nearly every published recipe. If you live in Europe, Fairy is the logical alternative and works just as well, as long as it's a concentrated version and not a "sensitive" or lower-surfactant one. The practical rule: don't chase the brand, chase the concentration — look for the "concentrated" or "ultra" version of either one, avoid anything labeled "sensitive skin" or "gentle" (they usually pack less active surfactant), and dial in the ratio with KUANTIKA's simulator based on your local humidity before you head out to test it.