In terms of vocabulary, Russian and Ukrainian share about 62% lexical similarity, which doesn’t necessarily mean that they have 62% of their vocabulary in common; rather, that 62% of their vocabulary are words with similar meaning and form and are from the same origin. For example, English and German share 60% lexical similarity.
This is considered a low percentage of lexical similarity for closely related languages. Ukrainian actually has a higher lexical similarity with other Slavic languages — it has 84% similarity with Belarusian, 70% with Polish, and 66% with Slovak.
Interesting article that talks about the similarities (most of the grammar) and differences (a large chunk of the vocab), as well as some of the historical reasons for the divergences (e.g. speakers finding themselves living in different countries following the 13th century fall of the Kievan Rus’).