How are 连A带B and A着B different
連A帶B is not equivalent to A着B.

A着B limits both A and B to verbs, but 連A帶B does not (they can be verbs, nouns, or adjectives).




Part of speech
Example sentence1
Translation




Verbs
眼下很多旅店為了拉客住宿都是連蒙帶騙。（1994年《人民日報》）
Many hostels nowadays (lit. 'blind' and) deceive customers for their stay.


Nouns
滅絕師太不除下劍鞘，連劍帶鞘，便向金花婆婆胸口點來。（金庸《倚天屠龍記》）
With the sword and its sheath intact, Madam Kill-All came for Golden Flower Granny's chest.


Adjectives
大夥也都連累帶餓快要頂不住勁兒了。（劉流《烈火金剛》）
Everyone was both tired and hungry and could not stand any longer.




連A帶B sometimes parts a compound verb (e.g., 連蒙帶騙–蒙騙, 連比帶劃–比劃, 連諷帶刺–諷刺), whereas parting them with 着 is nonsensical (i.e., it is wrong to say *蒙着騙, *比着劃, *諷着刺).


Swapping A and B in A着B is not always permissible.

There is 哭着說, but usually not *說着哭. We are modifying the main action (the second verb) with an adverb of manner (made from the first verb), so you can only say, 'He said something in the manner of crying', but he couldn't have cried in the manner of saying something. 說着笑 is not an appropriate example (by that, I mean the verbs in 笑着說 are not swappable, but 說着笑 does exist with a different meaning), because it can be the stative form of the compound verb 說笑 (to joke), which is something completely different and irrelevant to our discussion.

Swapping A and B in 連A帶B is not always permissible.

If there is no particular need for emphasis, no inherent logical order (邏輯順序), nor word order (e.g. existing compound words in point 2.), swapping is then permissible.




Example sentence1
Translation




母女倆用英文爭論，仁仁連手勢帶神色(=連神色帶手勢)都是美國式的。（嚴歌苓《花兒與少年》）
The mother and daughter argued in English; both Ren Ren's hand gesture and expression looked American.




Otherwise, swapping is not permissible.




Reason
Example sentence1
Translation




Emphasis on the first verb
「你沒走，這太好了，我都快急死了。」她連笑帶怨，發自內心的高興。（王朔《橡皮人》）
'Good, you didn't leave, you're driving me crazy!' She smiled with a tinge of complaint, her happiness from the bottom of her heart.


Logical order
某近視小姐，為同桌斟酒，竟連瓶帶蓋一起倒入杯中，⋯⋯（《市場報》1994 年）
A certain shortsighted lady was pouring wine for her table. She really poured into the cup all that was in the bottle, even the lid. (most of what was poured in was the wine in the bottle, not the lid)


Word order
所有的人聽到我叫喊都是連奔帶跑，只有羅非不肯動。（陸文夫《人之窩》）
All those who heard my shouting ran with all their might; only Luo Fei moved not an inch. (奔跑 is a set word; there is no 跑奔)




I think 連說帶笑 is more common than ?連笑帶說 for logical considerations: you first say something, and then laugh at what was said, not the other way round.


Reference:
1 肖奚強、余璐瑤（2017）〈“連/又/一邊A帶/又/一邊B”格式比較研究〉。南京師大學報(社會科學版)，6，137–146。