run about跑来跑去= run here and there

Put your coat on, or you'll catch cold running about.（美国英语常用run around,run about常用语英国英语。)

快把外套穿上,不然你到处乱跑会感冒的。

about

(1)prep. 关于；大约；在…周围 

结构A:动词+about+名词

(2)adv. 大约；在附近；在四周；几乎 

结构B:动词+about

(3)adj. 在附近的；四处走动的；在起作用的；在流行中的 

结构C:be+about+名词

about  adv.四下；到处(英文解释 If someone or something moves about, they keep moving in different directions.）

Everyone was running about. 

大家都在跑来跑去。

about  prep.围绕；环绕 (英文解释:If you put something about a person or thing, you put it around them.)  

Helen threw her arms about him.

海伦张开双臂拥抱他。

She was elegantly dressed with a double strand of pearls about her neck. 

她着装典雅，搭配着一条双层珍珠项链。

around

adv. 大约；旋转；到处，四处；在周围 

结构A:动词+about

prep. 围绕；在附近；前后，左右；在…周围 

结构B:动词+about+名词

around  prep.介词环绕；围绕；包围 （英文解释：To be positioned around a place or object means to surround it or be on all sides of it. To move around a place means to go along its edge, back to your starting point. 用于美国英语） 

She looked at the papers around her.

她看了看四周的文件。

Today she wore her hair down around her shoulders. 

今天她的头发披在肩上。

a prosperous suburb built around a new mosque. 

在新盖的清真寺周围建起的繁华市郊住宅区

adv.四处；在周围 

We have travelled around in Europe for six weeks.
我们在欧洲各地旅行了六个星期。

You have 15 minutes to look around.
你有15分钟的时间到周围看看。

a village with a rocky river, a ruined castle and hills all around.

有一条多石的河流、一座破败的城堡并且四面环山的村庄

The Memorial seems almost ugly, dominating the landscape for miles around. 那座纪念碑耸立在方圆数英里景致之中显得近乎难看。

    《牛津英语用法指南》第三版指出：around 和 about（美国英语中通常用 around）都可用来指不确定的移动或位置，表示“各处”、“到处”、“多处”、“某处”以及类似的意思。The children were running around/about everywhere. 孩子们到处跑来跑去。

这两个介词的“异同”，可以由后面的地点宾语来确定。

显而易见的是：run about everywhere = run around everywhere

但是，我们可以说：run around the lake  却不能说：run about the lake

鉴于上述区别，网友给的第2句，有两种意思。

The children are running about the playground.

孩子们在操场上跑来跑去。 

The children are running around the playground.

（1）孩子们在操场上跑来跑去。

（2）孩子们正绕着操场跑。