Is there full mutual intelligibility between Czech and Slovak language?

Czech and Slovak languages are indeed different. They developed independent from each other from Old Slovanic language. Czechs and Slovaks did not create common state until 1918. However, there was a huge influence of the Czech language over Slovak as Czechs had own universities and intelligentsia in the 19th century, while Slovaks were oppressed under Hungarians who prevented even publishing Slovak newspapers. Both languages are mutually understandable and they should not cause a issue with speaking and reading each other language. There are some vocabularies that weird and unknown within each language and must be explained, like medical terms or descriptions from history. For example Czech word for castle is hrad derived from old Slavic gradec (it became gorod in Russian), but Slovak word is kaštýl whose origin is Latin - castlellum and similar to word castle. Similarly Czech word for cemetery is hřbitov which is a German origin, but Slovak word is cintorín from Latin.

Czech language has characters that do not appear in Slovak and vice versa. Czechs have Ř which is unpronounceable to Slovaks (and for that matter to any Slavs), and ě. Slovaks have ä, ľ, and ô, which do not exist in Czech language. In general, the grammar rules between two nations are different and cannot be interchanged and it is hard to master by any non-native speaker. I can spot Slovakism in the Czech texts. Slovak grammar rule is easier as it was established later than Czech and they avoided pitfalls that plague the Czech grammar since the 19th century.

Generation that lived in common state of Czechs and Slovaks do not have problem to understand each other, but younger generation is already developing issues to grasp and understand the neighbor. This is nothing surprising as each nation develops its language. Czech and Polish language were understandable in one point in medieval times, but Czechs went through Reformation and Poland did not, which caused both languages to separate from each other. Czechs uses háček, which was product of the Bohemian reformation and it is not used in Polish language at all. Several generations from today, both nations will move to far from each other to make conversation between Czechs and Slovaks possible.