Doctrine means simply the formulated teachings of one's church about God, man, sin, immortality, etc. By contrast, dogma refers to a doctrine that we are told to accept by our religious authorities.
Dogmas are normative and authoritative doctrines. On this matter, there are significant differences between Protestants and other Christians.
For example, in Catholic circles, at least before Vatican II, dogma meant the decisions of all ecumenical councils from Nicaea in 325 A.D. on, plus all the “ex cathedra” utterances of every pope. For Protestants, dogma is a more hazy affair.
Some would say that the only Protestant dogma is that Jesus is the Christ. In Europe, even now, systematic theology is usually called dogmatics.
In English, however, the term 'dogmatic' has connotations of being narrow-minded, authoritarian, dictatorial, and arrogant, so theologians writing in the English language tend to avoid the term.