Looking at the competition between the USA and the USSR over the years, they seem primarily to focus their sports efforts in the Olympics in attempt to project each other's superiority during the Cold War.
However given the world prominence in soccer (football), why didn't they bother to develop strong teams to win the World Cup given the popularity of it around the world? Both the USA and USSR never won the World Cup or have really gotten close (excluding puppet or satellite states such as West Germany or Hungarian People's Republic).
Personally I feel like they put more investment into their respective Chess programs than in soccer...
However given the world prominence in soccer (football), why didn't they bother to develop strong teams to win the World Cup given the popularity of it around the world?
Why do you think that USSR didn't bothered? USSR soccer team won two Olympic torunaments (1956 and 1988), one European Championship (1960) and participated in 3 more finals (1964, 1972, 1988). Best result at World Cup - semi-final in 1966 (lost 2-1 to Germany).
They've "invested" and tried hard enough, just weren't good enough.
The Soviet football team was very good in the 1980s - consistently in the top 5, and at times first in the football Elo ratings. It was also good in the 1960s, with a weak interval in the 1970s.
The main reason that Russians are bad (for their population) at football is that the cold winters massively complicate training for a third to half the year. Hence why the Ukrainian team is about as good as the Russian one despite having a third of the population and with much less financing; and why the Caucasus countries (e.g. Georgia and Armenia) have much more footballing prowess per capita than does Russia, though small overall populations and poverty prevent them from being serious international contenders.