However
* The weather was terrible, however, the game went on as scheduled. (Contrasting two ideas)
* I tried to call you, however, I couldn't get through. (Introducing a contrasting idea)
* The project was a success, however, it was very challenging. (Highlighting a positive outcome despite difficulties)
Consequently
* The company lost a major client. Consequently, they had to lay off several employees. (Showing a result of a previous action)
* She studied hard for the exam. Consequently, she aced it. (Indicating a positive outcome due to effort)
* The roads were icy. Consequently, we decided to stay home. (Explaining a decision based on a situation)
Therefore
* The evidence is clear; therefore, the defendant is guilty. (Drawing a logical conclusion)
* The meeting was cancelled. Therefore, we can go to lunch instead. (Presenting a new course of action based on a previous event)
* The deadline is tomorrow. Therefore, we need to finish this project tonight. (Establishing a necessary action due to a deadline)