Neutrality is a snapshot, not a state. Any position with gamma stops being delta neutral as soon as the underlying moves, which is why the phrase always implies maintenance: you are neutral now, and you will not be in an hour.
Being delta neutral does not mean being risk free. The position still carries vega, theta and the second-order Greeks, and those are usually why it was put on. Neutrality just removes the directional bet so the other exposures can be traded cleanly.
Example: long ten XYZ $50 straddles with XYZ at $50.00. Net delta is roughly zero because the calls and puts offset. XYZ moves to $51 and the calls gain delta faster than the puts lose it, leaving the position long about 200 share-equivalents. Neutral lasted one dollar.
Related: delta-hedging, position-greeks, gamma, gamma-scalping