Russian is written in Cyrillic alphabet, which was invented in 9th century AD and was initially used for Old Church Slavonic (the oldest attested Slavic language, from a time when language of different Slavs was still mutually intelligible, and only later on evolved in separate languages). Cyrillic alphabet is derived form Greek alphabet. Latin alphabet is also derived from Greek alphabet, where Etruscan script was an intermediary.

So both alphabets derive from the same alphabet – the Greek one, and that's the reason for their similarity. There are different reasons for the cases where they differ, though:

1. For example English uses digraphs for sounds not found in Latin, for which as a consequence there are no letters (like sh, ch, th), some Slavic languages use modified letters in Latin script instead of digraphs (like Czech, Slovak, Slovene, Croatian š /pronounced as sh in ship/ vs s /pronounces as s in sip/). The Cyrillic alphabet approached such cases differently, the completely new letter was invented or taken from different script (for example letter for sh/š was taken from Hebrew) – in this case the letter was already borrowed from Hebrew in Glagolitic script (an older alphabet for Old Church Slavonic), and from there into Cyrillic.
2. The pronunciation of a Greek letter in Greek may be different through time, for example Greek beta was pronounced "b" when the Etruscans/Romans adopted the script and so the letter B is used in Latin script for the sound "b", in medieval Greek when Cyrillic was evolved it was pronounced as "v", and so in Cyrillic script the letter B is pronounced "v".

Added: I had to add the third reason, at first I thought it would be obvious, but it may not be so: 3) Greek influenced Latin and Cyrillic in different stages or types of its forms and through different intermediaries, so that's another reason for differences. For example, the Etruscan script evolved from some western type of Greek script, which had some peculiarities compared to what is now "standard" Greek alphabet, the letter lambda (Λ) was written differently, more similar to today's Latin L.