Strange errors occurring when compiling the 64-bit version of an application, error C2664

Sometimes you may see questions about strange errors generated by the compiler when building 64-bit code. A question may look in the following way:

//Class definition
class Type1 {...};
class Type2 {...};
class A
{
public:
...
void Func1(Type1* t1.....);
void Func1(Type2& t2.....);
...
};
//Using Func1 function
A obj;
Type2 t2;
...
obj.Func1(t2,...);
...

This code successfully compiles in the 32-bit mode but the compiler generates the error C2664 (Type2 cannot be cast to Type1*) when trying to build the 64-bit version. Although the function taking Type2& as the argument is defined, the compiler, due to some reason, tries to use the function taking Type1* as the argument. What is the matter?
Most likely, the problem is in the other parameters which were replaced by dots in the example. Here is one more example of the code:

class Type1 {};
class Type2 {};
class A
{
public:
void Func1(Type1* t1, unsigned &);
void Func1(Type2& t2, size_t &);
};
void use() {
  A obj;
  Type2 t2;
  unsigned u;
  obj.Func1(t2, u);
}

It successfully compiles in the 32-bit mode. But in the 64-bit mode the both functions fail to work. The compiler considers the first function a better candidate because its second parameter meets the condition. Yet it announces that the first argument does not suit: error C2664: 'void A::Func1(Type1 *,unsigned int &)' : cannot convert parameter 1 from 'Type2' to 'Type1 *'. The solution is to carefully study the other arguments and modify the code as needed.