A quick look on the history of the two clubs in terms of trophies:
FC Barcelona:
Domestic
Spanish Championship
Winners (21): 1928–29, 1944–45, 1947–48, 1948–49, 1951–52, 1952–53, 1958–59, 1959–60, 1973–74, 1984–85, 1990–91, 1991–92, 1992–93, 1993–94, 1997–98, 1998–99, 2004–05, 2005–06, 2008–09, 2009–10, 2010–11
Runners-up (22): 1929–30, 1945–46, 1953–54, 1954–55, 1955–56, 1961–62, 1963–64, 1966–67, 1967–68, 1970–71, 1972–73, 1975–76, 1976–77, 1977–78, 1981–82, 1985–86, 1986–87, 1988–89, 1996–97, 1999–00, 2003–04, 2006–07
Spanish Cup
Winners (25): 1909–10, 1911–12, 1912–13, 1918–19, 1921–22, 1924–25, 1925–26, 1927–28, 1941–42, 1950–51, 1951–52, 1952–53, 1956–57, 1958–59, 1962–63, 1967–68, 1970–71, 1977–78, 1980–81, 1982–83, 1987–88, 1989–90, 1996–97, 1997–98, 2008–09
Runners-up (10): 1902–03, 1918–19, 1931–32, 1935–36, 1953–54, 1973–74, 1983–84, 1985–86, 1995–96, 2010–11
Spanish League Cup
Winners (2): 1982–83, 1985–86
Spanish Super Cup
Winners (9): 1983, 1991, 1992, 1994, 1996, 2005, 2006, 2009, 2010
Runners-up (7): 1985, 1988, 1990, 1993, 1997, 1998, 1999
Copa Eva Duarte (Predecessor to the Supercopa de España)
Winners (4): 1945, 1948, 1952, 1953
Runners-up (2): 1949, 1951
European
European Cup / UEFA Champions League
Winners (3): 1991–92, 2005–06, 2008–09
Runners-up (3): 1960–61, 1985–86, 1993–94
UEFA Cup Winners' Cup
Winners (4): 1978–79, 1981–82, 1988–89, 1996–97
Runners-up (2): 1968–69, 1990–91
Inter-Cities Fairs Cup (the forerunner to the UEFA Europa League)
Winners (3): 1955–58, 1958–60, 1965–66
Runners-up (1): 1961–62
UEFA Super Cup
Winners (3): 1992, 1997, 2009
Runners-up (4): 1979, 1982, 1989, 2006
Worldwide
FIFA Club World Cup
Winners (1): 2009
Runners-up (1): 2006
FC Barcelona: Total number of trophies: 75 (61 national and 14 international)
Manchester United
Domestic
League First Division (until 1992) and Premier League: 19
1907–08, 1910–11, 1951–52, 1955–56, 1956–57, 1964–65, 1966–67, 1992–93, 1993–94, 1995–96, 1996–97, 1998–99, 1999–2000, 2000–01, 2002–03, 2006–07, 2007–08, 2008–09, 2010–11
Cups
FA Cup: 11
1908–09, 1947–48, 1962–63, 1976–77, 1982–83, 1984–85, 1989–90, 1993–94, 1995–96, 1998–99, 2003–04
League Cup: 4
1991–92, 2005–06, 2008–09, 2009–10
FA Charity/Community Shield: 18 (14 outright, 4 shared)
1908, 1911, 1952, 1956, 1957, 1965*, 1967*, 1977*, 1983, 1990*, 1993, 1994, 1996, 1997, 2003, 2007, 2008, 2010 (* shared)
European
European Cup/UEFA Champions League: 3
1967–68, 1998–99, 2007–08
UEFA Cup Winners' Cup: 1
1990–91
UEFA Super Cup: 1
1991
Worldwide
Intercontinental Cup: 1
1999
FIFA Club World Cup: 1
2008
Manchester United: Total number of trophies: 59 (52 national and 7 international)
Clearly we have a significant advantage. 16 trophies to be precise.