So, immediately after a finger amputation has occured, your first steps in terms of providing first aid are going to be ensure the scene is safe and that there are no additional life threats to the patient so. Life threats being things such as not breathing/no pulse and severe bleeding.
Now, for the finger... You're going to want to gently clean both the wound and the part that has been amputated with sterile water or sterile saline. It's not recommended on any of the resources I've looked at for the basic first-aider to be scrubbing out the wound or putting any heavy-duty antiseptics on either part at this point. Debriding the wound (scrubbing it) is best left to the medical professionals. You're going to then wrap the part that is still attached to the person in sterile gauze. You'll want to make sure that your bandage is not cutting off circulation since you still want good blood flow getting to that injury site for healing purposes.
Wrap the amputated part in a moist, non-fluffy sterile dressing, place it in a plastic bag and place that plastic bag on ice. I'd recommend double bagging this or sealing it in some other type of container. You do not want this ice and/or water getting in there and causing any type of freeze damage to that finger, nor do you want to risk that part becoming contaminated through the ice or water leaking into it.
You rinse both an amputated part and a tooth for the same reasons. To help remove any major debris as well as to give the part the cleanest environment you can prior to a doctor attempting any type of reattachment/reinsertion process.
I referenced Healthine's Severed Finger First Aid page, the American Academy of Orthopedic Surgeons page on Fingertip Injuries and Amputations and The American Association of Endodontists page on caring for knocked out teeth for this answer. I hope this helps and I welcome any questions or comments about any part of this answer!