The main difference that I can see (I don't have either machine, I've just looked at the specifications on your links) is that the Aldi PC has a separate graphics card, the Argos PC doesn't, it processes graphics on the motherboard.
This means that the Aldi PC will be better for games.
In general, Aldi PCs offer superb value for money. (I purchased one about 4 weeks ago).
However, they normally only have a few (10 max) per store.
Understandably, the staff (or their mates) quite often buy a few.
If you want one, you need to get to your Aldi store before 7am - there will be a queue - they will probably only have 4 or 5 available - they'll hand out tickets.
It's a lot less hassle to go for the Argos offering - especially if you're not a gamer.