There are a lot of factors and theoris that explain what happend to the Neardenthal in the end. The most common are:
1.- End of the fifth glacial age.
Homo Neanderthal, like the Homo Sapiens, based their life around hunting and recolection. But because of their specialise constitution (larger torax, thicker arms but shorter overall) and lack of flexibility in their survival practices, meant that, when the climate and ecosistem of Europe changed, they couldn't adapt (like most animals). They did have larger skulls, but smaller brains. Skull bones where thicker, better to bystand contutions, but restricted brain development. Is a shame, because they do had abstract thinking, a vital part of human development of creativity. This can be found in the expression of art and complex thinking like the concept of life after death, with buryals.
2.- Homo Sapiens arriving to Europe.
This one is pretty strigh forward. The interaction with homo sapiens, coming from Africa, drove away the neardenthal, forcing them to move their territory north. Thus, driven them extinct because of the harsher conditions and their innability to compete with the Homo Sapiens for reasources. This could been done through agresion for the better hunting grounds and water sources. Other reason is the interbreeding and aculturation of the Neardenthal hibrids into Homo Sapiens society and the aisolation of the Neardenthal, that has hindered them to adopt advance tecnology developed by the Homo Sapiens in the field of litic industy, superior to the Neardenthal's Musteriense tools (a tale as old as time). The interbreeding/aculturation proces is one of the strongets theoris right now. There is a lot of evidence from individuals of the period that present hibrid caracteristic. Nowadays, some modern humans has evidence of Neardenthal ancestry in their genotypes, that is present in their fenotypes. Like people with marked Supraorbital Torus, a Neardelthal caracteristic.