I've never dealt with either, but from everything I've read here, everyone says to go with Anker. Neither one will damage the phone, Anker just seems to put out a good product.
Fully charge (until the phone tells you to remove the charger) the battery, then use it (tiyr normal usage pattern) until the phone tells you to recharge it. Do that for 3 cycles (3 charges, 3 discharges), recharge it at around the 50% discharge point, and you'll be ready to toss the old phone (and it will be old) long before the battery gives out.