Is it actually helpful to write todo lists?
There is some research that suggests writing a to-do list will help "unclutter your brain." The research is related to the phenomenon known as the Zeigarnik effect. 

The basic observation behind the Zeigarnik effect is that we tend to think more about tasks and goals that are incomplete. This leads to, among other things, increased priming for concepts related to a unfinished goal relative to a finished one (Förster, Liberman, &amp; Higgins, 2005).

Masicampo &amp; Baumeister (2011) investigated whether making a plan to finish a goal, as opposed to actually finishing the goal, could lead to a similar reduction in intrusive thoughts about the goal. They report:


  In several studies, we activated unfulfilled goals
  and demonstrated persistent goal activation over time. Unfinished goals caused intrusive thoughts during
  an unrelated reading task (Studies 1 and 5B), high mental accessibility of goal-related words (Studies 2
  and 3), and poor performance on an unrelated anagram task (Study 4). Allowing participants to formulate
  specific plans for their unfulfilled goals eliminated the various activation and interference effects.
  Reduction of the effects was mediated by the earnestness of participants’ plans: Those who ultimately
  executed their plans were those who also exhibited no more intrusions (Study 4). Moreover, changes in
  goal-related emotions did not appear to be a necessary component of the observed cognitive effects
  (Studies 5A and 5B). Committing to a specific plan for a goal may therefore not only facilitate attainment
  of the goal but may also free cognitive resources for other pursuits. Once a plan is made, the drive to
  attain a goal is suspended—allowing goal-related cognitive activity to cease—and is resumed at the
  specified later time.


In sum, there seems to be some evidence that uncompleted tasks do lead to intrusive thoughts, and that finishing the tasks reduces intrusive thoughts. Alternatively, making and committing to a plan to finish the tasks can reduce the instrusive thoughts even in the absence of finishing the task.

References

Förster, J., Liberman, N., &amp; Higgins, E. T. (2005). Accessibility from active and fulfilled goals. Journal of Experimental Social Psychology, 41(3), 220-239.

Masicampo, E. J., &amp; Baumeister, R. F. (2011). Consider it done! Plan making can eliminate the cognitive effects of unfulfilled goals. Journal of Personality and Social Psychology, 101(4), 667.