The spell fear forces those affected by it to take the dash action on their turn. Can they do anything else besides that? Could a rogue dash with cunning action and still attack for example? Are they allowed to take back what they dropped when they failed the save against the spell?
Free your Action with Cunning Action (or some other Bonus Action Dash)
Fear Only specifies (emphasis mine):
a creature must take the Dash action and move away from you by the safest available route on each of its turns, unless there is nowhere to move.
This suggests that you must both move away and take the Dash action in doing so.
But the Rogue has a trick! They have a Cunning Action where they can use their Bonus Action to take the Dash action. This allows them to free up their Action for another use (attack, pick up a weapon, cast spell, interact with object, etc.)
Nowhere does it say that the individual under Fear must ONLY take the Dash action.
Drop it like it's hot
Fear also stipulates:
drop whatever it is holding
You have used your free object interation to drop your weapon at the time of the failed saving (not during your turn). It is now on the ground next to you until your turn (unless someone picks it up first.)
When your turn comes around, you now have to start moving with a Dash. However, you can pick up your weapon before you start moving with your no-cost object interaction (the turn penalty is only about moving away with Dash, which we've covered above.)
For those without an additional Action(Action Surge/Haste)/Bonus Action
They will have used up their Action to Dash and their movement. If they have an available Bonus Action, they may use that, but they are out of opportunities to pick their weapon (or whatever they're holding) back up.
The reason why they can't retrieve it on the same turn is that they have used their free object interaction to drop your weapon. If you want to pick it back up, you need to use an Action which is no longer available.
Those effected must take the dash action. That leaves a bonus and movement action for most characters.
Rogues are Cunning
A rogue can use cunning action to use their bonus action to take the dash action. This frees up their action for something else.
Starting at 2nd level, your quick thinking and agility allow you to move and act quickly. You can take a Bonus Action on each of your turns in Combat. This action can be used only to take the Dash, Disengage, or Hide action.
Pick Up Sticks
The character could use their move or action to pick up the dropped item as described in this answer