The new pull and the old pull both just feel like a pull, if you can only feel the pull at one spot. To see how the old pull is not like the new pull, you have to test the pull at a spot near you (but not the same spot), too. The new pull will be the same at each spot, but the old pull may not be the same (we call this the tide), and you test the sum of the new and old pull.
It's fair that it's hard to keep the two from becoming the same in your head, you need fancy stuff to test for the force of the tide, and they are more or less the same from a close-up (any which is much closer than, say, the moon) view!
