*** 片付けます・片付ける　is a transitive verb. e.g. 仕事 を 片付ける　= to get the job done; to complete the work 部屋 を 片付ける　= to put the room in order; to clean up the room *** 片付きます・片付く　is an intransitive verb.　 e.g. 仕事 が 片付く = The work gets done. The work is completed. 部屋 が 片付く= The room is put in order. The room is cleaned up.

It's just a matter of what is the subject. The one of the former is usually a person, the one of the latter is usually a thing.

It is a matter of preference, however, in my tentative opinion, I think Kanji-san's answer is more appropriate and useful in the end. Because the best answer here saying "It is all about the matter of the subject", explains "the effect" part of "cause-and-effect" while Kanji-san's explains "the cause" part. Why "the one of the former is usually a person"? It is because 片付ける is a transitive verb. And why "the one of the latter is usually a thing"? It is because "片付く“ is an intransitive verb. I understand that "simple conclusion without reason" style of the answer is sometimes more fascinating for leaners at a beginner level. I believe, however, it won't work in the end when you get into middle level. For example. ーーー　片づける お掃除ロボット（a machine/ a thing)が部屋をキレイに片付けてくれた！ 魔法のほうきがお部屋をキレイに片付けてくれた！ このひどい計画(a plan )が、俺達の夢をすべてどこかへ片付けてしまった。 　（It is a English-translation-like Japanese, but it is grammatical and works) or ーーー 片づく ウチの娘( a person) もやっと片付いたよ (Finally, she got married) Or you cannot understand 加える and 加わる with "thing-person subject" method. （くわえる） 私たち(person)は、あらたに一名、田中さんを新メンバーとして加えた。 （くわわる） 田中さん（person)が、新しいメンバーとして加わった。 But those lines can easily be understood by "transitive-intransitive" theory. I am sure the logic will take you further.

The former is a transitive verb and the latter is an intransitive verb.

Deleted and moved to Answers.