How to programmatically discover mapped network drives on system and their server names?

Have you tried to use WMI to do it?

using System;
using System.Management;
using System.Windows.Forms;

public static void Main()
{
    try
    {
        var searcher =  new ManagementObjectSearcher(
            "root\\CIMV2",
            "SELECT * FROM Win32_MappedLogicalDisk"); 

        foreach (ManagementObject queryObj in searcher.Get())
        {
            Console.WriteLine("-----------------------------------");
            Console.WriteLine("Win32_MappedLogicalDisk instance");
            Console.WriteLine("-----------------------------------");
            Console.WriteLine("Access: {0}", queryObj["Access"]);
            Console.WriteLine("Availability: {0}", queryObj["Availability"]);
            Console.WriteLine("BlockSize: {0}", queryObj["BlockSize"]);
            Console.WriteLine("Caption: {0}", queryObj["Caption"]);
            Console.WriteLine("Compressed: {0}", queryObj["Compressed"]);
            Console.WriteLine("ConfigManagerErrorCode: {0}", queryObj["ConfigManagerErrorCode"]);
            Console.WriteLine("ConfigManagerUserConfig: {0}", queryObj["ConfigManagerUserConfig"]);
            Console.WriteLine("CreationClassName: {0}", queryObj["CreationClassName"]);
            Console.WriteLine("Description: {0}", queryObj["Description"]);
            Console.WriteLine("DeviceID: {0}", queryObj["DeviceID"]);
            Console.WriteLine("ErrorCleared: {0}", queryObj["ErrorCleared"]);
            Console.WriteLine("ErrorDescription: {0}", queryObj["ErrorDescription"]);
            Console.WriteLine("ErrorMethodology: {0}", queryObj["ErrorMethodology"]);
            Console.WriteLine("FileSystem: {0}", queryObj["FileSystem"]);
            Console.WriteLine("FreeSpace: {0}", queryObj["FreeSpace"]);
            Console.WriteLine("InstallDate: {0}", queryObj["InstallDate"]);
            Console.WriteLine("LastErrorCode: {0}", queryObj["LastErrorCode"]);
            Console.WriteLine("MaximumComponentLength: {0}", queryObj["MaximumComponentLength"]);
            Console.WriteLine("Name: {0}", queryObj["Name"]);
            Console.WriteLine("NumberOfBlocks: {0}", queryObj["NumberOfBlocks"]);
            Console.WriteLine("PNPDeviceID: {0}", queryObj["PNPDeviceID"]);

            if(queryObj["PowerManagementCapabilities"] == null)
                Console.WriteLine("PowerManagementCapabilities: {0}", queryObj["PowerManagementCapabilities"]);
            else
            {
                UInt16[] arrPowerManagementCapabilities = (UInt16[])(queryObj["PowerManagementCapabilities"]);
                foreach (UInt16 arrValue in arrPowerManagementCapabilities)
                {
                    Console.WriteLine("PowerManagementCapabilities: {0}", arrValue);
                }
            }
            Console.WriteLine("PowerManagementSupported: {0}", queryObj["PowerManagementSupported"]);
            Console.WriteLine("ProviderName: {0}", queryObj["ProviderName"]);
            Console.WriteLine("Purpose: {0}", queryObj["Purpose"]);
            Console.WriteLine("QuotasDisabled: {0}", queryObj["QuotasDisabled"]);
            Console.WriteLine("QuotasIncomplete: {0}", queryObj["QuotasIncomplete"]);
            Console.WriteLine("QuotasRebuilding: {0}", queryObj["QuotasRebuilding"]);
            Console.WriteLine("SessionID: {0}", queryObj["SessionID"]);
            Console.WriteLine("Size: {0}", queryObj["Size"]);
            Console.WriteLine("Status: {0}", queryObj["Status"]);
            Console.WriteLine("StatusInfo: {0}", queryObj["StatusInfo"]);
            Console.WriteLine("SupportsDiskQuotas: {0}", queryObj["SupportsDiskQuotas"]);
            Console.WriteLine("SupportsFileBasedCompression: {0}", queryObj["SupportsFileBasedCompression"]);
            Console.WriteLine("SystemCreationClassName: {0}", queryObj["SystemCreationClassName"]);
            Console.WriteLine("SystemName: {0}", queryObj["SystemName"]);
            Console.WriteLine("VolumeName: {0}", queryObj["VolumeName"]);
            Console.WriteLine("VolumeSerialNumber: {0}", queryObj["VolumeSerialNumber"]);
        }
    }
    catch (ManagementException ex)
    {
        MessageBox.Show("An error occurred while querying for WMI data: " + ex.Message);
    }
}

to make it a little easier to get started download WMI Code Creater