Does the Melting Point or I²t (Ampere Squared Seconds) rating matter in a fuse? In short -yes, it can matter.
The purpose of this rating is to assure that the heat
created across the fuse during a surge has insufficient time to thermally conduct away
from the fuse to external circuitry. Once the measurements of
current (I) and time (t) are determined, it is a simple matter
to calculate melting I2t. When the melting phase reaches
completion, an electrical arc occurs immediately prior to
the “opening” of the fuse element.
In a fuse, you can have the same rated voltage, current, breaking capacity ratings along with being in the same response time category (slow, medium, fast) but have many different choices of the melting point.
Basically the higher the Melting Point (I2t) the longer the fuse will take to blow, or the more surges it can handle before opening.
This is different from breaking capacity or interrupt rating of a fuse, which is the max amount of current at a specified voltage the fuse can handle before rupturing or causing excessive arc. If you have ever seen cracked glass buss fuses, this means the breaking capacity of the fuse has been exceeded. Sometimes this can be worse and the fuse will shatter or explode if the breaking capacity is not properly selected. This can be a danger to surroundings, especially if there is flammable gas present.
Just like most electronic components, derating factors such as temperature should be applied also.
Further references are below: