Short answer, yes.
Many people have already answered the question, I just want to give you more information.
The 2 phrases are quite similar, and different people may have their own preference using the words. To make it simple, you can consider they are 99.99% the "same" and can be used interchangeably. On daily usage, "打扫房间" is more common than "清理房间", but I guess it is regional dependent.
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Let's focus on the words 清理 and 打扫.
清理
- dirt/rubbish removal
- remove anything that is unwanted ( to free up space and tidy up the room )
In another word, 清理房间 imply room cleaning by removing rubbish / dirt and tidy up the room. It also means to throw away unwanted items to free up space.
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打扫
扫 is actually 빗자루 in the verb form. 打 is an action you pat on something to remove the dust. Of course, in modern days, we actually use a vacuum cleaner and other cleaning tools.
In another word, 打扫房间 simply means room cleaning and nothing more.
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However, in reality, we also throw away things and tidy up the room during 打扫房间. that make "打扫房间" and "清理房间" no difference!
( Modern Chinese is very flexible )
The Korean phrase 청소 하다 is actually equivalent to the traditional 한자 ---> 淸掃 / 淸扫, but in modern Chinese, we no longer use the old word phrase 淸掃 / 淸扫.
In Cantonese, a single word can be used to describe the action -- "執“ .
清理房间 / 打扫房间 in Cantonese is 執房