Sánchez attended high school in Santurce. After graduation, he attended Ohio State University where he graduated with a degree in engineering in 1934. He then was a professor for a short time at the University of Puerto Rico.
During his tenure, Sánchez tried to change his party's membership, urging a younger generation to rise in the party's organization. It could be argued that Sánchez was influenced by the youth movement that the island was experiencing countrywide during the 1960s, a period where many social areas in Puerto Rico, including television, music and sports, were being introduced to fresh, younger personalities.
Sánchez had public marital problems during his term; he divorced his wife in 1967 and married one of his close assistants, Jeannette Ramos-Buonomo, the daughter of the then Speaker of the House, Ernesto Ramos Antonini. This marked the first time a Governor married while holding the office. His marital problems, which would have not played a public role in later years (judging by the public reaction to Sila Calderón's divorce and remarriage during her administration in the early 2000s), were brought to center stage during the 1968 gubernatorial campaign, given the still conservative Puerto Rican morals of the time.
His goals of revitalization and change led to a public break with former governor, Luis Muñoz Marín, who was still party leader. Because of this, the PPD nominated Luis Negrón López for Governor in the elections of 1968. Sánchez Vilella left the PPD and founded his own party, the short-lived Partido del Pueblo (The People's Party). While his new party lost in the 1968 elections, it caused a considerable percentage of PPD voters to vote for him, indirectly helping Luis A. Ferré and his New Progressive Party to an upset win that year. It did not matter that Ferré had made a similar split with his Partido Estadista Republicano (Statehood Republican Party) due to that party's decision not to support the statehood status option in the 1967 Puerto Rican status plebiscite (Ferré created the New Progressive Party of Puerto Rico as a consequence). There is even speculation that United States Navy officials, scorned by Sánchez as long-time meddlers in Puerto Rican affairs (in a by-now famous episode, Sánchez had thrashed a hotline telephone that linked the governor's office and the local Navy command) used Navy intelligence resources to prepare a smear campaign against Sánchez and helped Ferré with logistics and money for his own gubernatorial campaign. Nevertheless, Sánchez was finally blamed for the first loss in the history of the PPD. His relationship with former governor Muñoz Marín was severely strained, but the two friends mended their differences in the late 1970s.
In 1972, Sanchez made his third and last run for elective office when he obtained over 100,000 votes in his bid to become a representative-at-large, but lost when the Supreme Court of Puerto Rico certified the election of Puerto Rican Independence Party (PIP) candidate Luis Angel Torres, who polled fewer than 150 votes, based on its interpretation of the Puerto Rico Constitution's rules regarding the election of at-large legislative candidates.
In 1997, Governor Pedro Rossello signed into law a bill introduced by then senator Kenneth McClintock converting a major highway built by Sanchez Vilella between Ponce and Mayagüez into the "Roberto Sanchez Vilella Expressway", honoring not only his service as Governor but as Secretary of Public Works.
The legacy of Sánchez Vilella, who was initially judged rather harshly by historians, has been perceived in a better light recently. His term was overshadowed by the legacy of Muñoz, his predecessor, and the PDP's loss in the 1968 elections is still blamed on him. However, he is perceived by many political commentators to have led the most efficient public administration of all Puerto Rican-born governors, and many long for Sánchez's frankness and political integrity in light of the deteriorating political climate that has developed after he left office. As the perception of his legacy improves, he is now being honored more frequently. The Government's largest building complex at Minillas in Santurce, has now been named after him, and Senate President Kenneth McClintock placed a bust of Sánchez Vilella in 2007 at the Capitol Building's Governors' Hall, righting a decades-long omission.
His son Roberto Sánchez Ramos was appointed by Governor Sila Calderón as Solicitor General and was narrowly confirmed by a coalition of PDP and Pedro Rosselló loyalists in the opposition-led Puerto Rico Senate as Secretary of Justice of Puerto Rico in 2005, after being appointed by Governor Aníbal Acevedo Vilá.