Differences between template specialization and overloading for functions?

So, I know that there is a difference between these two tidbits of code:

template <typename T>
T inc(const T& t)
{
    return t + 1;
}

template <>
int inc(const int& t)
{
    return t + 1;
}

and

template <typename T>
T inc(const T& t)
{
    return t + 1;
}

int inc(const int& t)
{
    return t + 1;
}

I am confused as to what the functional differences between these two are. Can someone show some situations where these snippits act differently from each other?