LIBOR is the price of money in the UK, ironically the most internationally open financial market while simultaneously a tightly regulated domestic financial market. This rate is sampled from all member banks and is what they are charged by other banks to borrow. LIBOR covers many currencies and maturities.
The US Prime rate most commonly refers to the Wall Street Journal's Prime Rate which is "The base rate on corporate loans posted by at least 70% of the 10 largest U.S. banks." Different countries have different prime rates.
As one can see, it's difficult to associate the two because one is an international rate charged to international banks while the other is a "base rate" charged to corporations in the United States though both are for USD.
The WSJ Prime appears more stable because 7 out of 10 reporting banks must have changed their reported rates for the overall rate itself to change while LIBOR takes an average of the middle 55% reported values daily. Also, the interest rates of USD in the US are based somewhat on US inflation and mostly on the Fed Funds Rate especially at the short end, so they will be more stable than the world USD inflation rate which is ignored by the Federal Reserve and subject collectively to each country's demand for USD credit.