When can modified duration and effective duration be close to each other?

They can be close for an option-free fixed-rate bond when cash flows are fixed, the rate move is small, and the effective-duration model is using a parallel benchmark shift that is economically similar to the bond's yield change.

The important word is "can." Modified duration assumes cash flows do not change. Effective duration is based on model prices after up-rate and down-rate shocks. For a plain fixed-cash-flow bond, those approaches may line up. For a callable bond, putable bond, or mortgage-backed security, expected cash flows may change when rates move, so effective duration is usually more appropriate.