In Dutch, the Dutch language becomes "Nederlands", and the Dutch people become "Nederlanders". However, in German, the German language becomes "Deutsch" and the German people are "Deutschen". Why we don't call the Dutch "Netherlanders" and the German "Dutch"?
This is, from my understanding, a result of the use of the word "duitsen" in Dutch to describe themselves, for example in Het Wilhelmus, in the first stanza, the line "Ben ick van Duytschen bloed", which, in modern Dutch, is "ben ik, van Duitsen bloed". While today the word "duitsen" is actually a plural feminine for "German", it previously referred to anyone of German/Dutch descent/blood.
During the time which "duitsen" and its Middle Dutch counterpart "dütsch" were in use in the Netherlands, the word entered the English language, and became "Dutch".
German, on the other hand, is from Latin, "germanus", meaning a "person from Germany". I am not, however, familiar with the reason English uses "German" over French "Allemande" or an Old English equivalent, such as "þeodisc".
To elaborate a bit on /u/mousefire55's answer as it applies to English-language usage of the words, it's useful to keep in mind that the modern concepts of "German" and "Germany" aren't much older than the country itself. Before that, words like "Dutch", "German", and "Teutonic" referred quite broadly to what today are thought of as dialects of German and Dutch, and sometimes even to all Germanic languages: apparently Scottish chroniclers writing in Latin referred to the Scots language (which is closely related to English) as "Teutonicus". While there were of course distinct subregions of this large Germanic-speaking area, the relevant ones weren't modern national borders but smaller duchies or principalities like Friesland or Bavaria.
When the Netherlands emerged as a political entity in the 16th century, the people who lived there were naturally called "Dutch", as were their Germanic-speaking neighbors in what is now Germany, Austria, Switzerland, Luxembourg, etc. If you needed to refer to them specifically, you could say "low Dutch" ("Netherlands" just means "low countries"); there wasn't a word then specifically for what we'd call Germans, because there wasn't any country with those borders.
This situation was still in place when large numbers of German-speakers settled in Pennsylvania, which is why their descendants (and the language some of those descendants still speak) are called "Pennsylvania Dutch".
Things changed when most of the German-speaking states began the process of uniting into a single country with an emperor. As this country came to be called "Germany", the word "Dutch" was used increasingly to refer only to the Netherlands and its languages, rather than to the entire language family and region.
Hi, there's room for more depth on this, but for now you can get started on this earlier post