Multinomial and Poisson are very different. Multinomial regression should be used when your response is categorical with more than 2 categories. Poisson regression is used when your response is a count of an incident.
Classic Poisson regression: response is # of accidents at an intersection in a day. This can be modeled using Poisson regression since there is no obvious upper bound to the # of accidents. Also note that the numbers actually mean something and have quantitative value.
Classic example of multinomial regression: response is dietary habits of an individual, Non-vegetarian, Vegetarian or Vegan. Note that the variable in qualitative and not necessarily quantitative. And there are restrictions to what values the response can take (1 of three).
There are almost no obvious scenarios that I can think of where these two methods can be used interchangeably. For count data, if you want other options, you might look at Negative-Binomial Regression.
If the goal is to model the effect of the treatments, where the effect can be categorical (got well, stayed same, got better) a multinomial (or logistic) regression should be used with the treatment as a predictor.