Heeren House is a more homey "B&B" style place which is in an ideal spot right next to the river (such as it is!). Puri is a bit further up the street but still in the old part of town.
For value for money, I am afraid I have defected to Puri as I thought Heeren was quite pricey last time we stayed there. If you book on line you can get a pretty good room rate. www.hotelpuri.com.my (I think).