詰める and 包む: putting inside a container and wrapping the outside
Compare packing clothes inside a suitcase with wrapping a gift in paper. Read the container marked by に and the material marked by で without assuming a completely full container.
Put the items inside a container
詰める, read つめる, means pack or reduce a gap. 旅行かばんに着替えを詰めた。 says that a change of clothes was packed in the suitcase.
旅行かばんに identifies the container receiving the clothes. Packing is reported, but a completely full suitcase with no gaps is not established by this sentence.
Wrap the outside of an object
包む, read つつむ, means wrap or envelop. 贈り物を青い紙で包む。 says that someone wraps or will wrap the gift in blue paper.
青い紙で identifies the wrapping material. Distinguish putting something inside a suitcase from wrapping its outside in paper, and keep 包む separate from a past completed action.
A different use for reducing a gap
席を少し詰めれば、もう一人座れる。 says that if people sit a little closer together, one more person can sit. 詰めれば is a condition and 座れる a possibility.
詰める also describes reducing a gap. Do not force the selected inside-packing versus outside-wrapping comparison onto every use of the words.
Does 旅行かばんに着替えを詰めた。 necessarily mean the suitcase is completely full with no gaps?
Show the answer and why
No. It reports putting the clothes inside the suitcase. The remaining space or degree of fullness needs additional information.