My pen tool isn't snapping to anchor points

I agree with S_Gans. This usually happens is Snap-to-Point is set.

Definitely make sure Smart Guides are on by going to ... View > Smart Guides and ensure it is checked

If you find the snapping annoying in other projects, you can also turn Snap to Point off by going to ... View > Snap to Point

May I assume you have your Preferences set to Snap To Point, in the Selection and Anchor Display area?

If so, I recommend turning on Smart Guides, and only clicking when you are hovering and see the word "anchor" appear.

One other thing - if you're drawing with the pen tool, and click an existing anchor point, the tool will automatically assume you're intending to delete it if it's in the middle of a path, and it will assume you want to continue drawing FROM it if you click the end of an open path.

If you simply want to attach a new drawing, using a pre-existing anchor point, I recommend creating a new sub-layer, and locking the sublayer on which your original point exists.