As a native speaker of German, I have to admit that I find both sentences slightly odd. Especially the second sentence makes no sense. I was wondering from which old-fashioned book they came until I read on and saw that the grammar book is from 1968. This explains a lot. I would rewrite the sentences as follows:
(1) Anna spielt gerne Klavier, aber nicht so gut wie ihre Mutter. (Anna likes playing the piano, but doesn't play as well as her mother.)
(2) Seine Stimme ist nicht stark, aber auch nicht zu schwach; sie ist schoen. (His voice is not strong, but not too weak either; it is nice.)
You are right that there are coordinating and subordinating conjunctions, but this only determines the position of the verb in the sub-clause. Coordinating conjunctions (aber, oder, und , denn, doch, sondern) keep the original sentence structure, where as subordinating conjuctions invert the position of the pronoun and the verb. Examples for subordinating conjuctions are:
allerdings (however/ indeed), als ob (as if), als (when), bevor (before), bis (until/ by the time), da (since/ because), daher/ darum/ deshalb (therefore/ that's why), damit (so that), daß (that), falls (if), indem (by), nachdem (after), ob (whether/ if), obwohl (although), seit/ seitdem (since), sobald (as soon as), sooft (as often as), soviel (as much as), soweit (as far as), trotz (despite), trotzdem (nevertheless), während (while), weil (because), wenn (when)
The position of the conjunction is more flexible in German than in English, for example:
Er würde gerne kommen, aber er kann nicht kommen. (He would like to come, but he can't come.)
Er würde gerne kommen, er kann aber nicht kommen! (He would like to come, but he is really not able to come!)
By moving "aber" to another position in the sentence, we change the intensity of the meaning, which is only possible by adding "really" to the English sentence and by using "able to" instead of "can".
I hope this helps!
Viel Spass weiterhin mit Deutsch!