Both great bikes, reasonable spec for the price, but slightly different in intent and purpose.
The anthem s a short travel XC bike. It'll be fast over rough stuff, climb well, and descend reasonably well. It has a steepish head angle for a full susser, which will make it pretty fast but hard work on steep descents. It has good shock, decent fork, but Giant' policy is to sell everything ont he same frame, so its ripe for upgrades when the OE wears out. The review of the next bike up is very complementary from Bike Radar, and suggests this could be made a lot lighter with such upgrades.mforthe budget though, this doesn't look bad.
It'll be ideal for fast singletrack and most trail centre riding as well as average Bridleway stuff. 100mm travel will be fine for that sort of thing, if speed and smooth ish stuff is your bag.
The Trance is a different beast. It has 140mm travel; more than enough for the average red trail centre, and will eat most black graded stuff for breakfast. It has a slack head angle of 67 degrees, which will make it a good prospect for steep downhills. It'll climb reasonably well with its 73.5 degree seat angle, and the frame and shocks will soak up some pretty big hits. Giants are, in my opinion good looking things. I love the frame designs, but I am very wary of the overdrive steerers They effectively limit you to Giant specific forks, although their stiffness and precision are not in question.
So. If you want to do some hairy trail riding and rough technical stuff, the 140 bike is the answer. If you want to climb like a goat, thrash along the flat and mince down the rocky stuff, then the Anthem is the answer.
As an out and out trail rider, rather than a whippet, I'd choose the 140 bike.
Have you looked at any other brands? I'd be tempted to look at Norco ( from
Evans) or Lapierre, Canyon, Cube... Plus many more, although decent spec at the same sort of money will be limited if you look at say Specialized etc.