Thanks to Rob Kennedy for pointing me in the direction of ShGetFileInfo. I then Googled on that and found these two examples - Delphi 3000, Torry's. From that I wrote the following class to do what I needed.
Also, just as I was finishing up Bill Miller's answer gave me the final bit of help I needed. Originally I was passing full file names through to ShGetFileInfo, which wasn't ideally what I wanted. The tweak suggested of passing "*.EXT" was great. 
The class could do with more work but it does what I need. It seems to handle file extensions that have no details associated either. 
Finally, in what I'm using I've switched it to using a TcxImageList instead of a TImageList, since I was having problems with black borders appearing on the icons, because it was a quick fix.
unit FileAssociationDetails;
{
  Created       : 2009-05-07
  Description   : Class to get file type description and icons.
                  * Extensions and Descriptions are held in a TStringLists.
                  * Icons are stored in a TImageList.
                  Assumption is all lists are in same order.
}
interface
uses Classes, Controls;
type
  TFileAssociationDetails = class(TObject)
  private
    FImages : TImageList;
    FExtensions : TStringList;
    FDescriptions : TStringList;
  public
    constructor Create;
    destructor Destroy; override;
    procedure AddFile(FileName : string);
    procedure AddExtension(Extension : string);    
    procedure Clear;    
    procedure GetFileIconsAndDescriptions;
    property Images : TImageList read FImages;
    property Extensions : TStringList read FExtensions;
    property Descriptions : TStringList read FDescriptions;
  end;
implementation
uses SysUtils, ShellAPI, Graphics, Windows;
{ TFileAssociationDetails }
constructor TFileAssociationDetails.Create;
begin
  try
    inherited;
    FExtensions := TStringList.Create;
    FExtensions.Sorted := true;
    FDescriptions := TStringList.Create;
    FImages := TImageList.Create(nil);
  except
  end;
end;
destructor TFileAssociationDetails.Destroy;
begin
  try
    FExtensions.Free;
    FDescriptions.Free;
    FImages.Free;
  finally
    inherited;
  end;
end;
procedure TFileAssociationDetails.AddFile(FileName: string);
begin
  AddExtension(ExtractFileExt(FileName));
end;
procedure TFileAssociationDetails.AddExtension(Extension : string);
begin
  Extension := UpperCase(Extension);
  if (Trim(Extension) <> '') and
     (FExtensions.IndexOf(Extension) = -1) then
    FExtensions.Add(Extension);
end;
procedure TFileAssociationDetails.Clear;
begin
  FExtensions.Clear;
end;
procedure TFileAssociationDetails.GetFileIconsAndDescriptions;
var
  Icon: TIcon;
  iCount : integer;
  Extension : string;
  FileInfo : SHFILEINFO; 
begin
  FImages.Clear;
  FDescriptions.Clear;
  Icon := TIcon.Create;
  try
    // Loop through all stored extensions and retrieve relevant info
    for iCount := 0 to FExtensions.Count - 1 do
    begin
      Extension := '*' + FExtensions.Strings[iCount];
      // Get description type
      SHGetFileInfo(PChar(Extension),
                    FILE_ATTRIBUTE_NORMAL,
                    FileInfo,
                    SizeOf(FileInfo),
                    SHGFI_TYPENAME or SHGFI_USEFILEATTRIBUTES
                    );
      FDescriptions.Add(FileInfo.szTypeName);
      // Get icon and copy into ImageList
      SHGetFileInfo(PChar(Extension),
                    FILE_ATTRIBUTE_NORMAL,
                    FileInfo,
                    SizeOf(FileInfo),
                    SHGFI_ICON or SHGFI_SMALLICON or
                    SHGFI_SYSICONINDEX or SHGFI_USEFILEATTRIBUTES
                    );
      Icon.Handle := FileInfo.hIcon;
      FImages.AddIcon(Icon);
    end;
  finally
    Icon.Free;
  end;
end;
end.
Also here is an example test app using it, it's very simple, just a form with a TPageControl on it. My actual use was not for this, but for with a Developer Express TcxImageComboxBox in a TcxGrid.
unit Main;
{
  Created       : 2009-05-07
  Description   : Test app for TFileAssociationDetails.
}
interface
uses
  Windows, Forms, FileAssociationDetails, Classes, Controls, ComCtrls;
type
  TfmTest = class(TForm)
    PageControl1: TPageControl;
    procedure FormShow(Sender: TObject);
    procedure FormClose(Sender: TObject; var Action: TCloseAction);
  private
    { Private declarations }
    FFileDetails : TFileAssociationDetails;
  public
    { Public declarations }
  end;
var
  fmTest: TfmTest;
implementation
{$R *.dfm}
procedure TfmTest.FormShow(Sender: TObject);
var
  iCount : integer;
  NewTab : TTabSheet;
begin
  FFileDetails := TFileAssociationDetails.Create;
  FFileDetails.AddFile('C:\Documents and Settings\...\Test.XLS');
  FFileDetails.AddExtension('.zip');
  FFileDetails.AddExtension('.pdf');
  FFileDetails.AddExtension('.pas');
  FFileDetails.AddExtension('.XML');
  FFileDetails.AddExtension('.poo');
  FFileDetails.GetFileIconsAndDescriptions;
  PageControl1.Images := FFileDetails.Images;
  for iCount := 0 to FFileDetails.Descriptions.Count - 1 do
  begin
    NewTab := TTabSheet.Create(PageControl1);
    NewTab.PageControl := PageControl1;
    NewTab.Caption := FFileDetails.Descriptions.Strings[iCount];
    NewTab.ImageIndex := iCount;
  end;
end;
procedure TfmTest.FormClose(Sender: TObject; var Action: TCloseAction);
begin
  PageControl1.Images := nil;
  FFileDetails.Free;
end;
end.
Thanks everyone for your answers!