Essentially a firqa is a major difference in belief and a madhab is set of differences in practice.
For example although Salafis are unlike Sunnis in practice they are far closer than Ahmadis to Sunnis in belief. On the other hand Ahmadis may even follow the madhab of Imam Abu Hanifah, but because their beliefs differ they are hence called a firqa.
It is important that farq should be assessed by what people say about themselves not about what others say about them.