In the Brønsted–Lowry model, an acid can donate a proton and a base can accept one. The same substance can sometimes behave differently depending on what it reacts with.
Where do we see it?
In water, pH gives a practical measure related to hydrogen-ion activity. A change of one pH unit represents a tenfold change in this activity, so the scale is logarithmic rather than linear.
Neutralisation does not always mean the final solution is exactly pH 7. The result depends on the acid, the base, their amounts and the chemistry of the products formed.
pH is not the whole story
Two solutions with the same pH can behave differently depending on the species present and their total concentration. pH is useful, but it is not a complete description.
Go one step further
A useful way to understand this phenomenon is to separate the driving force from the visible result. Temperature, pressure, concentration, surface area and the nature of the materials can each change the rate or the final state. In real systems, several of these factors often act at the same time.
That is why the same phenomenon can look different in a laboratory, in the kitchen or outdoors. The underlying chemistry or physics stays the same, but the conditions change the balance.