Ideological differences
The sheer volume of nuclear missiles that had been amassed by each side meant that a nuclear apocalypse was possible. The Crisis would not have become so intense if so much had not been at stake.
Nuclear posturing
US spy planes spotted the construction of missile bases on Cuba. These were capable of launching ICBMs (intercontinental ballistic missiles). Soviet nuclear weapons were now in America’s ‘backyard’.
This led the Americans to assume that the Russians had altered their thinking from subscribing to the MAD (Mutually Assured Destruction) theory to a ‘first strike’ policy.
Khrushchev’s decision to place nuclear weapons on Cuba can be seen as an attempt to counteract NATO's Jupiter missiles in Turkey.
These missiles were a similar distance from the USSR as Cuba was to the USA, and had the potential to strike anywhere in the USSR.