This topic will always be an interesting topic to discuss about. Judging from the recently ended English Premier League season, Man Utd can pride themselves and say that they are the better team because they won the league. But when you want to compare these two teams, we need to compare the amount of trophies won in each of the club's history. In terms of major trophies, Liverpool are still way ahead of Man Utd. Man Utd only holds the edge in the English FA Cup count. Liverpool are ahead 18 as compared to 16 in terms of the league, 5 to 2 in terms of the European Cup, 3 to 0 in the UEFA Cup, and 7 to 2 in the English League Cup. Man Utd only holds the edge in terms of the FA Cup with a 11 to 7 record. So, Liverpool has won 40 major trophies in it's distinguished history with Man Utd winning 27 major trophies. In short, Man Utd are 13 trophies short to match Liverpool's tag as the most successful British football club in history. It is definitely possible for Man Utd to overtake Liverpool but it could take a good part of two more decades.
One distinct difference between Liverpool and Manchester United are it's fans. Liverpool fans are basically the best fans in the world. There is no other fans in the world who would actually cheer louder when their team is losing or lost a match. The recent European Cup final in Athens was a clear example. The sound of You'll Never Walk Alone was deafening. They even cheered and gave AC Milan a standing ovation for their deserved victory. That is special. The atmosphere at Anfield is absolutely out of this world. Man Utd may have their Theater of Dreams at Old Trafford but Anfield is definitely the Theater of Passion.
Besides that, these 2 teams are amongst the most successful clubs in world football. It would be an never ending debate to discuss which team is actually better. Many football forums would have this topic and it will be flooded in no time. But, in terms of the trophy count and the fans, Liverpool FC are undoubtedly the more successful team at the moment. Period.