Detergents are designed to remove stains, grease and other particles. They are typically natural desiccants and very abrasive. Most detergents also contain a vast number of chemicals, many of which may be poisonous. The chemical makeup of the detergent will vary between brands and the detergent's function, whether it be for washing clothes or washing dishes, but their detrimental natures remain largely the same.
Mixing detergent in with a snapdragon's water will inevitably stunt its growth, kill it if it has already bloomed, or prevent it from blooming entirely. The abrasive nature of the detergent will strip away at the plant's waxy coating, which will make is susceptible to various diseases. The poisons in the detergents may also prevent it from taking in nutrients, meaning less energy for growth and blooming.
Adding detergent water to the snapdragon's soil creates a toxic environment. Many detergents contain large amounts of sodium, particularly those that are labeled as "softening" detergents. This sodium builds up in the soil and prevents the snapdragon's roots from taking in enough nutrients. The sodium also soaks up water that the plant would otherwise use, further retarding the snapdragon's growth.
Though few people would ever mix detergent in with the water they use for their snapdragons, it is possible that the chemicals reach the plants via gray water. Gray water is water that has been run through washing cycles, like in the washing machine or dishwasher, and is laced with the detergent as well as other particles. Some may use this gray water for their plants as a mean to save on water bills, but over time this gray water will be detrimental to your snapdragon's health.