Let’s see what we can find out about this question, shall we?
Of course, Put[a]in and the Puttanescas will repeat the Tsarist trope that Ukrainian and Belarusian are mere dialects of Russian, while Ukrainian nationalists in particular will vehemently assert an independent linguistic identity. Of course, by the “Max Weinreich rule” (“a language is a dialect with an army and a navy”), all three East Slavic tongues qualify as separate languages. But what about more objective linguistic criteria?
For example, how mutually intelligible are the languages? In the case of Ukrainian and Russian, that is asymmetric: almost every Ukrainian understands Russian, but the opposite is not the case. This sort of asymmetry also exists between German and my native Dutch, by the way: Most Dutch speakers can sort-of figure out German (I read and understand standard German fluently), but fewer Germans (except from the border provinces where Plattdeutsch/Low German is still in some use) will understand Dutch.
Another criterion is linguistic distance: assembling a core vocabulary corpus (e.g., with the Dolgopolsky list or one of the Swadesh lists) for both languages involved, then measuring the lexical distances between corresponding word pairs (e.g., by the Levenshtein distance) and adding them up.
This appears to be what was done (for Dolgopolsky) in a map by an Ukrainian linguist named Tishchenko that has been making the rounds of the net. (English translation with modifications by Alternative Transportation blogger Stephan Steinbach:.)
If we zoom in on the Slavic cluster:
That Ukrainian and Polish are actually closer than Ukrainian and Russian… makes sense from a historical perspective (5-6 centuries of being under Polish[-Lithuanian] rule). So does the high degree of similarity between Ukrainian and Belarusian. (The dissimilarity index isn’t printed here, but a straight unbroken line means 25 or less.)
For comparison and perspective, let’s zoom in on the Germanic cluster, with which I’m most familiar:
So it would seem that in this map, Ukrainian and Russian are about as close as Dutch and English, while Ukrainian and Polish are a bit more distant than Dutch and German.
It appears from the work of Maurizio Serva of the University of Rome “La Sapienza”, on the phylogenesis of Indo-European languages, that if one uses one of the Swaresh lists (which are longer), Russian and Ukrainian again become closer.
If one makes the corpus really large, one might come to counterintuitive conclusions. For example, if one analyzes the 85,000 entries of the Shorter Oxford English Dictionary, one would conclude English is a Romance language, since Old French and Latin together account for a plurality of word origins. However, if one filters down to the 1,000 most commonly used words, 85% of those are Germanic.
Regardless of whether Ukrainian and Belarusian are closer to Polish or to Russian, or about equidistant: there is no doubt the two languages were very significantly influenced by the West Slavic branch. In contrast, Russian was more strongly influenced by the South Slavic branch, specifically by Old Church Slavonic, the liturgical language of the Russian Orthodox Church. Later, of course (from the days of Peter the Great on), German and French (and even some nautical Dutch!) loanwords entered the Russian language, but this also happened in Ukrainian (sometimes filtered through Polish).
Outside the Indo-European family, Russian absorbed some Tatar and other Turkic loanwords during the days of the Golden Horde, while Ukrainian picked up some Hungarian through contact in its most western region.
PS: This paper:
M. Willems, E. Lord, L. Laforest, G. Labelle, F. J. Lapointe, A. M. Di Sciullo, V. Makarenkov, Using hybridization networks to retrace the evolution of Indo-European languages, BioMedCentral Evolutionary Biology 2016, 16, 1–18. http://doi.org/10.1186/s12862-016-0745-6
goes a step further than the Levenshtein distance model in that it tries to also describe “cross-fertilization” between the languages as their evolved.