The US' racism problem on steroids.
Martin Luther King joined the Nation of Islam instead and took up Malcolm X's cause of violent black nationalism. Malcolm X became the only strong black voice and millions of African Americans over decades converted to Islam. This created a perfect platform of hatred for the devoutly Christian, white supremacist Ku Klux Klan, who portrayed the violent black Muslims as a double threat - being black and carrying the Sword of Islam. Funnily enough, JFK being assassinated really helped civil rights by bringing in the experienced Southerner, Lyndon B. Johnson, as president to pass the Civil Rights Act - so JFK isn't killed for some reason in this timeline and fails catastrophically in his efforts.
Many riots, bombings and top-level assassinations by black and white supremacists led to the creation of the new Sudanese Empire (derived from the Arabic term of Sub-Saharan Africa, the land of the blacks) in the South in majority-black areas, causing the far larger KKK of TTL to declare the secession of the Confederate States and thus starting the Second American Civil War. The federal government broke down as its legitimacy was wiped out, many states seceded and declared their neutrality or backed the Confederacy in the 2nd ACW. That pretty much counts as a collapse of the US.
For the Soviet Union, with no huge American threat, the relationship between Moscow and Beijing massively sours until border skirmishes turn into a full out war. The inexperienced and backwards Chinese army slowly give way to the Soviets, but not without huge losses. When two countries relied on their vast manpower in mass human offensives, millions were sure to die for a few kilometres of land. The USSR could never occupy China wholly, as a foreign force, so it planned to set up a puppet government afterwards. For the cost of hundreds of millions, it replaced a communist dictator with another one? Safe to say, people weren't happy about the loss of most of their army and men in a seemingly pointless war. The rebellions and unrest could be easily crushed, but the economic cost of the war and occupation broke the Soviet Bear's back. Much like Afghanistan did in OTL (but on a far far far larger scale), it was a huge drain of money for a communist dictatorship to sink into, and by the time the USSR pulled out, its economy was in shambles. A military coup took control of the Kremlin, causing the USSR to fragment into a mess of warlords as the rule of law broke down. Ethnic minorities quietly formed their own states, either in a somewhat democratic fashion or a military dictatorship. At least in a civil war, the nations are in a formal state of war but in the former USSR the regimes barely even control a tenth of the land they actually claim. China has also fragmented into a mess of warlords and military juntas after the Soviets pulled out. The mess of the USSR and China afterwards makes the collapse of the US look like an orderly rule-bound sporting match.