In many cases, you can normalize both of the Unicode characters to a certain normalization form before comparing them, and they should be able to match. Of course, which normalization form you need to use depends on the characters themselves; just because they look alike doesn't necessarily mean they represent the same character. You also need to consider if it's appropriate for your use case — see Jukka K. Korpela's comment.
For this particular situation, if you refer to the links in Tony's answer, you'll see that the table for U+00B5 says:
Decomposition <compat> GREEK SMALL LETTER MU (U+03BC)
This means U+00B5, the second character in your original comparison, can be decomposed to U+03BC, the first character.
So you'll normalize the characters using full compatibility decomposition, with the normalization forms KC or KD. Here's a quick example I wrote up to demonstrate:
using System;
using System.Text;
class Program
{
static void Main(string[] args)
{
char first = 'μ';
char second = 'µ';
// Technically you only need to normalize U+00B5 to obtain U+03BC, but
// if you're unsure which character is which, you can safely normalize both
string firstNormalized = first.ToString().Normalize(NormalizationForm.FormKD);
string secondNormalized = second.ToString().Normalize(NormalizationForm.FormKD);
Console.WriteLine(first.Equals(second)); // False
Console.WriteLine(firstNormalized.Equals(secondNormalized)); // True
}
}
For details on Unicode normalization and the different normalization forms refer to System.Text.NormalizationForm
and the Unicode spec.