As some people have noticed, both words exist in Russian, buy they have different connotations. According to linguists and sociologists, when people use the word Fatherland (отечество) as it has masculine gender, they refer to the government structures, army etc, it is kind of rational. So, "Fatherland" exists as long as there is a country to defend, generally speaking. "Motherland" (родина) has feminine gender, so it is associated with your family, memories, emotions or something that a particular person values. People often refer to it as "Родина-мать" ("motherland-mother" which is a tautology in English), so it is associated with a mother tongue, the place where you were born and nature (according to the sociological survey).

The concepts of Motherland is wider than Fatherland, so Motherland includes it.

Edit: by the way, the world "Motherland" is not exclusively Russian, you can find it in many Slavic languages, such as Czech, Ukranian or Slovak "rodina", Polish "rodzina" or Bolgarian "родзiна".

Edit 2: as people in comments below correctly pointed, you can find the same roots in other Slavic languages meaning "family", not "motherland". My bad, I should have checked that before including in the edit.

The term 'fatherland' meaning the 'land of (fore)fathers' is very common and may stem from the Greek 'patrida' or Latin 'patria'. Many languages, including German, use various forms of this term and thus it is one of the most popular ways to denote a native country. It should also be noted that in English, both 'Fatherland' and 'Motherland' are used, with the latter first attested in English in 'The Tragedie of Gorboduc' by Thomas Norton and Thomas Sackville published in 1561 with Oxford Etymological Dictionary citing its origin to the Middle French term 'terre mere' already present in 1480s. The term 'fatherland' is older, as the Middle English 'fæderland' meant 'inheritance from father' (cf. e.g. Polish 'ojcowizna') and in its modern form and meaning it has been recorded around 1620, as a loanword from German or Dutch.

In German, the term 'Vaterland' is first attested in early 12th century text 'Summarium Henrici' where it also means 'the inherited land', although the meaning 'land of the ancestors' appears already in the late Middle Ages. It became widely used in the mid-19th century in the wake of the nationalistic ideas appearing throughout Europe that in German states were also corresponding with the notions of national unification. The term 'Vaterland' was used in the 'Song of Germany' written by August von Fallersleben in 1841 for the Haydn's melody of the 'Gott erhalte Franz den Kaiser', a birthday anthem in honour of the Emperor Francis II. The word gained popularity in the process of the unification that led to the formation of the German Empire and became firmly associated with Germany during the Great War and Second World War with the 'Vaterland' being commonly used in the propaganda posters and public speeches, especially in the Third Reich. Today, the connotation is so strong that in common parlance the terms are rarely used in Germany, with the equally old word 'Heimat' ('homeland') being used instead.

The case of Russia is a bit more complex, as in Russian there are two words used to denote the subject of this question. First is 'отечество' (otechestvo) or 'отчизна' (otchizna), derived from Russian 'отец' (atets) meaning 'father' and thus a direct equivalent to Latin 'partia' (a noun referring to something related to father(s), although this noun is of neutral gender). The other is 'родина' (rodina), a feminine noun derived from 'родить' ('to give birth' or, in reciprocal voice, 'to be born') making it closer to an English word 'homeland'. On a side note, this word in West Slavic languages (pol. 'rodzina', cz./sl. 'rodina') means 'family'.

So, why the term 'Motherland' is commonly associated with Russia in English? As we're speaking about linguistic usus rather than rules that are set in stone, we can base this on several sound conjectures referring to the personification of Russia throughout history. The first instances of the country being portrayed as a motherly figure (Mother Rus, then Mother Russia) come from at least 15th century and is attested in works of an Orthodox scholar Maxim the Greek and letters of Andrey Kurbsky (sometimes rendered as 'Krupsky'), commander in the employ of Ivan IV. The association is quite natural, as both Rus and Russia are feminine nouns in Slavic languages. The actual origin is hard to ascertain, some point to the influence of the cult of Slavic fertility goddess Mokosh, although as very little is known of the pre-Christian Slavic culture due to extreme scarcity of sources, this is tenuous at best. The term 'Mother Russia' seems to predate the term 'otechestvo' that went into widespread use only during the reign of Peter I (early 18th century), very possibly due to strong German cultural influences in the wake of Westernization attempts by the tzar, although it didn't displace the common imagining of the 'Mother Russia'. The motherly personification, as common among the royalists, has been initially discarded by the communists during the October Revolution, but since the late 1920s and return of the more nationalist approach to politics, the idea of "Mother Russia' being a benevolent parental figure to all other Soviet republics gained popularity. This was evident in the well-known propaganda poster by Herakliy Toyidze 'Родина-мать зовёт!' (lit. 'Country-mother calls!') made when the Germany attacked USSR in 1941 and less known 'За Родину-мать!' (lit. 'For the country-mother!') from 1943. This personification was also used for the monument 'Ро́дина-мать зовёт' on the Mamay Kurgan commemorating soldiers fallen in the Battle of Stalingrad (now Volgograd, Russia) that merges the image of Nike with that of Mother Russia. Popularity of such imagery was the general reason why the Russia is called 'Motherland' in English.

Hi!

Not to discourage answers, but you may find these threads useful:

https://www.reddit.com/r/AskHistorians/comments/1kosud/why_are_some_countries_popularly_depicted/
https://www.reddit.com/r/AskHistorians/comments/1ps9lx/why_did_russians_in_ww2_refer_to_russia_as_the/
https://www.reddit.com/r/AskHistorians/comments/19bjzd/fatherland_vs_motherland/