'Take into account' is used in contexts where an entity is regarded/considered/given thought for an operation/action.
Eg: A football club must take into account the fans' feelings
before selling off their star player.
'Account for' refers to provisioning for an entity's involvement in an action/operation.
Eg: A good football club must account for player salaries when
preparing their annual budget.
The key difference is the tone implied by the two phrases. When you 'take something into account' you take it into consideration while performing an action.
If you have to 'account for something', you have to make clear provisions for it during your action.
Let's look at the following to sentences:
1. A football club must take into account the fans' feelings before selling off their star player.
2. A good football club must account for fans' feelings before selling off their star player.
Sentence 1 implies that a football club must give thought to the fans' feelings before selling their star player. The club should consider the fans' emotions and morale before they go ahead with the player sale.
Sentence 2 implies that a good football club must make provisions for the damage incurred by fans' feelings due to the sale. The club could either make another star signing or reduce ticket rates or do something to account for the fans' feelings.