Some clarification in this question is needed here. Are you asking why the Soviet Union did not create its own automobile industry or are you inquiring about the automobile industry of the Russian federation after the breakup of the USSR?

The answers to either question are going to be very different and in the latter case, bump up against the twenty-year rule a little.

I don't have time right now to write a post, but this is a good book about Soviet car industry: Cars for Comrades: The Life of the Soviet Automobile by Lewis H. Siegelbaum

To be short - lack of experience, technology, control and car industry for a long time wasn't a priority, because an idea of personal cars was "capitalistic".

"A huge totalitarian system has been brought to its knees because nobody wants to wear Bulgarian shoes." -- P.J. O'Rourke

or

The Gold Plated Ambulance

In short, the Russians were always able to build cars, but never enough and never ones that people with options wanted to buy. The car industry is hyper competitive and consumers expect tangible improvements every year. Without a top end to the market with substantial disposable income underwriting R&D efforts Russian car makers had little incentive to create innovative improvements. Things that are now standard in most cars, anti-lock brakes, airbags, power windows, etc, were all pioneered as very expensive features on luxury brands. As they were refined and the cost to produce them reduced they filtered down into the normal market. There is no mechanism for this in a centrally planned economy. After the fall of the Soviet Union car makers lacked both the organization expertise and the capital to upgrade facilities in order to compete with the West and the Japanese and Koreans.

You could also argue that the European nations had been industrialized quite a long time before Russia. USA also had the benefit of investments from Europe. And in Europe there where countries which had large empires abroad. If we take these factors into account they where at a disadvantage from the start. Furthermore the US didn't suffer as much from the second world war as Russia. The industries in west Germany also received quite a lot of investments from the Marshall plan.

Another point that could be made is that even though Russia have had excellent military engineering they've often created simpler and cheaper alternatives. The obvious example would be the ak-47. So the general tendency would be to create simple durable products. So it all boils down to how we measure greatness in cars. The Lada Niva is quite popular as a terrain vehicle since it's known to be reliable. Which can prove very helpful in some conditions.