The difference is mostly the unit composition. There is only one book for all german "branches", which allows you to create a fallschirmjäger or SS or conventional army.
Overall though the SS and fallschirmjäger are mostly just more experienced infantrymen, who are harder to kill, have moral boost (especially SS). I I remember correctly those units also have access to a wider choie and number in advanced weapons, but I haven't got my book at hand right now.
I've never played against the british, but as far as artillery is concerned, they do pack a lot'a punch. However, the main issue with artillery at that point isn't that it hurt since officer activated ordance needs you to roll a 6 to actually inflict hits, but the ludicrous amount of pins it can spread over your whole army. Considering the reduced size of your army, a single unit not acting a turn because they're pinned down is a real pain. The best advice I can give you is to spread your forces out to not end up with haf your footmen hitting the dirt, and pray for the barrage to have bad rolls...
As far as tanks are concerned there's no real difference, simply put them veteran if you want to bring along an elite themed army. This essentially boosts "moral".