The two terms are basically synonyms, but there are some differences.
First, it's not merely a matter of "age." Karl Barth wrote Church Dogmatics in 1932, while Charles Hodge wrote Systematic Theology in 1872.
On the other hand, there seems to be a geographical bias. Barth, Berkouwer, Heppe, Pieper, and Ott all wrote originally in German or Dutch, and their works are translated into English with a variation of the word "dogmatic." Heppe is a good example: his Reformierte Dogmatik became Reformed Dogmatics.
There are exceptions: English writers Edward A. Litton (Introduction to Dogmatic Theology, 1882) and William Shedd (Dogmatic Theology, 1889) eschewed "systematic," even though their peers Boyce, Dabney, Miley, and Hodge preferred it.
Some attempt to draw a strong distinction in the meaning of the terms, arguing that "Dogmatic" implies a closer relation with the teachings of schools and denominations, and less of an emphasis on the Bible, than is implied by "Systematic." For example, the Grace Institute for Biblical Leadership puts it simply:
Systematic theology [...] uses the Bible as its primary source. [...] Dogmatic theology uses as its primary sources the creeds or statements of faith.
John Miley does not paint so stark a picture, describing the difference and similarities as follows:
Dogmatic theology has its proper distinction from both biblical and systematic, though often used in the same sense as the latter. It is not limited to the Scriptures, like the biblical, nor has it by any requirement the comprehensiveness of the systematic. [....] Mostly, dogmatic theology devotes itself to the creed of a particular school. [...] [It] may be just as free from dogmatism in any philosophic sense of the term, and just as scientific in its principles and method, as systematic theology. (Systematic Theology, vol. 1, Introduction)
One author particularly involved in the discussion is Louis Berkhof, as his treatise was first published as Reformed Dogmatics in 1932, and then republished several years later as Systematic Theology (1938). The preface of a modern publication of that work states:
[Berkhof] yielded to the suggestion, perhaps from his publisher, that Reformed Dogmatics was not a term and title as clear to the American readership as Systematic Theology -- although he continued to view the original title as more fully descriptive of his purpose.
Berkhof makes his case for the term Dogmatics in an early chapter:
Reformed scholars in Germany and in the Netherlands show a decided preference for the title Dogmatics, with or without a modifier. In our own country, however, the term Systematic Theology seems to have a more popular appeal. From an ideal point of view the former certainly deserves preference, (1) because it is the more specific of the two, and designates the real object of study with greater precision; and (2) because the modifier 'systematic' in 'Systematic Theology' is apt to create the impression that the study under consideration is the only theological study which treats its subject matter in a logical order.
But he does not consider this a crucial distinction:
For practical reasons, however, it seems more desirable, especially in our country and in our day, to use the title Systematic Theology. This does not require the sacrifice of any principle.
In fact, Berkhof notes that Benjamin B. Warfield (Presbyterian and Reformed Review, 7-26 (1896)) prefers Systematic. Warfield disputes Berkhof's second point, writing:
What is meant by calling this discipline "Systematic Theology" is not that it deals with its material in a systematic or methodical way, and the other disciplines do not; but that it presents its material in the form of a system.
In Berkhof's definition of Dogmatic Theology from a reformed perspective, he like Miley points out its tie to a particular system of doctrine but emphasizes the importance of scriptural backing:
It seeks to give a systematic presentation of all the doctrinal truths of the Christian religion. [...] The dogmatician must demonstrate the truth of the system which he presents as his own. He must show that every part of it strikes its roots deep down into the subsoil of Scripture.
It thus seems safe to say that in general, the terms can be viewed as near synonyms, with minor, technical distinctions, and that their usage depends more on geography, language, and personal preference than a universally accepted difference in meaning.