Generally whenever there is a conviction on any driving violation, its appearance on your driving record will affect your car insurance rates. Your rates will generally increase, but by how much it is difficult to say.
Washington's Department of Licensing keeps a record of all traffic infractions, convictions, and notices for failures to appear or respond to traffic citations. The courts forward all of this information to the DOL. Once the DOL receives notice of an entry of a conviction, points are placed on your driving record (5 points for Negligent Driving, First Degree).
While under deferred prosecution, you have not been convicted of anything, so you have had no jail time or any driver's license suspension - although you are generally driving under a probationary license for 5 years during this deferred prosecution period.
The DOL will base its actions on your license at the time it receives the entry of conviction from the court. If your violation was originally charged as a DUI, this stays on your record for 15 years, however the action on your license will be based on the convicted charge. When the court amends a charge, as in your case from a DUI to Negligent Driving, the court will strike out the original charge and add the new violation code and write "amended" on the citation or complaint.
Any action on your driver's license would be taken when the DOL receives this amended complaint and conviction. Regardless of the deferred prosecution, you may still have a driver's license restriction or suspension based on the Negligent Driving charge and the date of the conviction - these driver's license actions are set by statute (RCW 46.20.270) and may not be deleted. In other words, even with your satisfactory completion of probation and the reduction of your charge from a DUI to Negligent Driving, you may still find that the DOL will take license actions (restrictions, suspensions, revocations) based on this entered conviction.
Again, I am by no means an expert in Washington driving laws, but if I am reading the Washington Revised Code and the Washington DMV information correctly, this is what you can expect when your probationary period ends and the Negligent Driving conviction is entered by the court.