**Heads-up: this is a quick partial answer — please accept a more detailed one.**

-----

### It’s about omitting postpositions/markers.

- All of the sentences sound good and natural to me.
- Grammatically, both are identical. Not this one, but _there are some cases where fuller forms are not idiomatic_ and sounds more awkward. E.g. “너 심부름 가니?” sounds much better than “너 심부름을 가니?”. The exact reasons and rules ⸺ I don’t know.
- Although _those without postpositions are more common and seemingly preferred_ by native speakers, it is not that you can drop any postpositions and make sense. From a related paper:
>
> ### Omitting agentive postpositions (subject markers)
>
> **TL; DR: if a subject is specific and always interpreted as a subject (“never misinterpreted as an object”) even without a subject marker, you can drop the marker.**
>
> ‘너 심부름 가니?/저 개 아주 사납다/OOO의 소설 재미있더라.’와 같이, 대명사나 그 지시 대상이 특정적인, 한정적 표현의 명사가 주어 자리에 올 때 주어가 주격 조사 없이 쓰일 수 있다.
> When a subject is a pronoun or a specific noun, you can omit the agentive postposition (subject marker).
> - ‘**너** 심부름 가니?’ “Are _**you**_ going on an errand?”
> - ‘**저** 개 아주 사납다.’ “_**That** dog_ is so aggressive.”
> - ‘**……의** 소설 재미있더라.’ “_**……’s** novel_ was interesting to read.”
>
> 하지만 ‘개 아주 사납구나./소설 재미있더라.’와 같이, 주어 자리에 오는 명사가 비한정적일 때에 주격 조사가 쓰이지 않으면 비문(非文)이 된다.
> If you don’t put an agentive postposition (subject marker) on a subject that is not specific, the sentence becomes ungrammatical.
>
> 또한 ‘그 여자 무얼 주려고 하는데.’와 같이 주격 조사가 없이 쓰인 체언이 주어 이외의 것(‘그 여자가’ 혹은 ‘그 여자한테’)으로 혼동될 우려가 있을 때는 주어 자리에 오는 체언이 비록 한정적이라도 주격 조사가 쓰여야 한다.
> If a subject without an agentive postposition (subject marker) could be misinterpreted as a non-subject, you have to keep the agentive postposition (subject marker), whether the subject is specific or not.
> - The ambiguous sentence ‘그 여자 무얼 주려고 하는데.’ (“The girl; about to give something.”) can be interpreted as:
>   1. ‘그 여자(가)<sub>[subject “She”]</sub> (누구에게)<sub>[implied object “to someone”]</sub> 무얼 주려고 하는데.’ “She(’s) about to give something (to someone).”
>   2. ‘(내가)<sub>[implied subject “I”]</sub> 그 여자(한테)<sub>[object “to her”]</sub> 무얼 주려고 하는데.’ “(I’m) about to give something (to) her.”
>
> ⸻ Gisim Nam, ‘현대 국어 통사론’ (“The syntax of modern Korean”), published by 태학사 (Taehaksa), in 2001.

Some other postpositions you could omit (couldn’t find documented rules though):
- 목적격 조사<sub>Accusative postposition (object marker)</sub> “을”/“를”
- 속격 조사<sub>Genitive postposition “’s”</sub> “의”
- 향격 조사<sub>Lative postpostion “to”</sub> “에”/“로”
- and possibly more.