Villa's motives for the Columbus raid seem to have been various. He wanted to take revenge on the American arms dealer who had taken money for supplies and neither given him the arms or returned the money. He was increasingly angry at the way the US professed to be neutral in the Revolution, but actually aided Carranza. He also hoped to gain arms and supplies. He achieved none of these. Instead, instead of easily overcoming what he'd been told was a garrison of 50 US soldiers , he encountered one of 600. Instead of opening fire in a surprise attack on the barracks, Villa's troops opened fire on the stables. As a result of a vigorous defense and counter-attack, Villa lost men- especially officers- and was forced to retreat back to Mexico empty handed.
The Punitive Expedition was ordered by President Wilson after Villa's raid, intended to find Villa and his army. About 5,000 soldiers, it had the initial tacit cooperation of Carranza, who helped supply it by railroad. Despite the advantages of force and supplies, it was however trying to operate in Chihuahua, Villa's home: and so Villa knew the territory. Chihuahua was/is also the biggest state in Mexico, sparsely settled and with rough terrain, so capable of hiding armies. There was a great deal of local resentment to having a US army operating in Mexico, despite Pershing being under strict orders to treat civilians with great respect. Pershing soon discovered that no one would supply his army with information about Villa's whereabouts, and rightly suspected that the locals were also constantly supplying Villa with information about Pershing's. Villa also discovered that Carranza's troops were also often more sympathetic to his army than to the Americans, and would sometimes look the other way when his men were in an area. When Villa was wounded in the knee, he essentially dispersed his units and hid in a cave for two months. This also briefly created the impression that he was possibly dead, and the Expedition therefore successful.
Pershing had had some experience in fighting guerrillas in the Philippines and pretty quickly assessed the situation, saying that a much longer time would be needed, and more resources, if Villa was to be found. But as the Expedition continued, Carranza became himself more hostile to the US, more hostile to having the US Army on Mexican territory and less cooperative, and his troops began to show some real armed resistance to the US, for example blocking them at Parral. Things might have escalated further, but neither Wilson or Carranza wanted to enter into a war: Mexico was too divided and weak, and the US was anticipating being pulled into WWI in Europe for which it would need an entire army. After about a year of futility, Pershing was put back into a more defensive position along the border, and eventually all US forces left Mexican soil.