as you can clearly see - and as i observed in one of my rare sp-games in tournament -  the hard setting simply means that your teammate ai sucks ass and feeds the opposing ai and that is what makes it actually "hard"

hard and nightmare have increased health regen, gold income, and experience rates. they also use a somewhat smarter AI than the easy or normal comps but as far as i can tell hard and nightmare are equally smart, just that nightmare gets WAY more bonus stuff. 

The AI is generally very poor at managing flag control and citadel upgrades, both as an ally and as an enemy.  Enemy AI units typically buy priests and angels at inappropriate times (ie. as soon as possible) and feed you EXP/gold, and allied AI often do nothing with the citadel, though I have noticed that friendly AI will often upgrade to Giants if I have already bought up to Catapultasaurai (I hate typing that).  Dominating the flags (and therefore war score) and then upgrading direct to CpSa or Giants midgame is generally the best way to beat the hard and nightmare AIs.  I haven't had the outleveling problem against Hard (and that is as Erebus, a DG who often has leveling problems early game), but Nightmare will often outlevel me, in addition to cheating.  Overall, though, if you're having trouble beating even the Nightmare AI, a human team is going to do unpleasant things to your anus.

On prison, yes, flag control even against the dumb AI is difficult on nightmare because it's a fairly "open" field.  On crucible, I disagree.  Very often the AI will not make strategically sound decisions regarding the two flags on the "peninsula" in the middle.  If you're good and not playing a weak one-on-one character (thinking specifically of TB), it is fairly easy to defend the path onto the peninsula and dominate the two flags past it.  I've had games where, through ample use of TP scrolls, I never lost control of them after the first cap.  If you're really a rook player (as you were in the screenshot), a good tower farm on that lane is particularly devastating on Crucible, and you can always make excursions to cap the central flag, as well.  This also works well against humans.  Allowing Rook to set up and hold a tower farm on the peninsula in Crucible is basically fatal.

Flag control and clever citadel management is the biggest advantage you have over the AI.  Learn to use those aspects of the game to your advantage, and you'll be rolling the AI in no time.  Out-think them; don't out-level them.

Nightmare AI gets x4 cash, not x2.  Try playing a "filthy rich" game on Prison (or another very small map) then kamikaze on the fortress with Oak for the win.  You'll have just over 25k gold, while the nightmare AI will have a whopping 100k.  It's virtually impossible to beat nightmares on high cash settings as a result, since you really need to build up a long-term advantage.

Here is my little advices to improve playing Demigod:

DONT PLAY VS THE AIS!!! you'll get bad habits and in the end you'll be slaughtered online vs serious players!!!

However if you really really want to here is my suggested training:

Play Prison map with conquest and all normal setting and do the following schedule:

a) Pick a demigod you like and want to improve and do a 1vs1 vs a normal ai

once you win consistently without suffering any deaths play 1vs1 vs 1 hard ai

c) once you win consistently play alone vs 2 normal ais, that should teach you map control.. repeat till you win confidently without suffering any kill

d) Do the same as c versus 2 hard ais

e) Now you can beat easily 2 hard ais, you're no longer a complete newbie and can be of some use.. now you can start tacking on 1 nightmare ais, you'll see that unless you're quick and play aggressively ais tend to gain level much faster than you and with lot more money...

f) You now beat the 1 nightmare ais like a cakewalk, you're ready to take vs 2 nightmare ais

g) Congratulations, you can now win vs2 nightmare ais, repeat till you win it without dying a single time

h) By this point you should know how to kill the opponent interrupting his fleeing attemp, you should be careful not to be ambushed and not dying stupidly and you should have a decent map control strategy you're now ready to go online and be a decent/average player

If now you can either try with another demigod or going out fightin vs 3 nightmare ais

U'll also find out that some demigods ais are more annoying and difficult to beat than others, that depends also about your choices... for me one of the most annoying on a 2vs1 match are unclean beast and regulus but thats just my observations.

At any point in the schedule if you want you can try to put an hard ai on your side in which your task will be to keep him alive and assist him as much as possible, that will teach you how to help dumb allies that playing witthout any self-preservation usually end up losing the game for your team.

Have fun and good luck!