It is important because the Sun follows the popular mood, so if the Sun has decided the popular mood is to ditch Labour, then they are probably right.
I don't think the Sun is that influential as to deliver tons of votes to the Conservatives that they would not get anyway.
Unlike say the Daily Mail which will never support Labour, the Sun is the floating voter of Fleet Street. If the Sun is changing sides, then it is following the national trend rather than setting it.
They usually claim "it was the Sun wot won it", but that is a boast without any truth.