On first thought I'd say Demo because the weapons have a higher potential damage (50+50+75 vs 50+50+50) and Soldier and Demo have equal start health (200).

Soldier might have a slight strategic advantage because the shotgun has a double-shot, but demo is still stronger overall.

I'd say terrain matters a lot. Demo gets trapped easier because digging up with drill is very difficult. I've even attempted dynamite jump with a trapped demo. Worked like once.
Soldier can climb/escape girder traps with shotgun blasts more easily.
Also demo only has one ranged source of damage. Soldier has 3.

You know, solution is to modify The Specialists with a script param that defines specialist order per team ☺
Like...
S=(S)oldier D=(D)emo E=(E)ngineer N=(N)inja P=(P)yro C=(C)lown H=(H)olyman[saint] X=Sniper[running out of letters, but X-out or something]
t1=SENPCHXD,t2=DPNEHCXS

Then play a 1 v 1 battle to 7 v 7 without sharing soldier or demo.

Any undefined team would get standard order.

Could maybe alow repeats too... t1=SSEENNDD

I'm not a pro at Hedgewars, so the long-range combat that the soldier is key for doesn't exactly fit my combat style, and wind tends to be your enemy at the worst moments. I will admit that the shotgun is definitely helpful during battle, but I prefer the demo's arsenal over the soldier's

Plus, I like things that blow up and have satisfying explosions, which the demo allows with his dynamite

Demo supporter all the way!

BTW, I went ahead and added the customisation, so this sort of stuff can be playtested now.
Using the game scheme config script param line:
t1=XXXXXXXN,t2=HC would make t1 only snipers apart from a last ninja, and t2 start with saint and clown

t1=SEE
replaces team #1's ninja with another engineer.

Could playtest these scenarios with complete teams of 8 or just 1v1 specialist battle.

It's on DLC as The Specialists (Custom) and I checked it in for next release as a standard config option.

Oh yeah!!! Thx nemo
I will playtest this stuff as soon as I can!