Solo queue is the game mode for people who like to play cancer champs like Kalista or Vayne and don't want to risk their desired champ being banned.
Team builder is a longer wait but safer to get a reliable team composition. You can leave a queue when you see a Twisted Fate ADC with a Cho'gath support and a Kassadin jungle and not get penalized for it with a wait time. There's actually a very short wait for anyone who wants the support role which 90% is what I end up wanting. One of the downsides is that you often end up with a rag tag group of people while the other team is consistently a group of IRL friends using teamspeak who have an advantage over you.
Draft is nicer. There is more time to communicate, you can plan to remove cancerous champs from a game and you can actually pick in order to counter an enemy's pick. There's more strategy than sheer luck involved when picking your champs.