They are more or less the same.
However, there are some differences:
1. 死に絶える is a more colloquial expression than 絶滅する.
絶滅する is a more formal and academic word.
2. 絶滅する is usually used for a species. In a figurative talk, it can be used for more widely.
死に絶える is used for a family, a race or a species.
明智小吾郎には孫がおらず、名探偵の血筋は死に絶えた。 All his "family" members were dead.
三葉虫 (the trilobites) は絶滅した。
三葉虫は死に絶えた。 You may use this expression in a poem or lyrics, but 三葉虫 and 絶滅した has a better collocation because you may see 三葉虫 in academic articles much more frequently than in romantic poems.
Hope this helps.