From Mac OS X version 10.6 onwards, Apple insists that we buy iLife for exploring photos from the iPhone. In previous versions of OS X, I could easily explore my iPhone photos, using iPhoto directly and can easily copy.
But newer versions of Mac OS X force us to buy iLife.
Is there any alternative for exploring photos of iPhone like iPhoto in OS X?
You actually don't need to download anything extra at all. Apple includes an application called Image Capture as part of the default OS X install. You can use it to import pictures from your camera/iPhone, as well as scanning in photos from a scanner.
Your question is unclear for a few reasons. First of all, iLife is not an app, it's a package that includes iPhoto, iWeb, iMovie, and Garage Band. Also, iPhoto is allowed and in fact is the preferred method of managing photos on your iPhone. There's nothing about Snow Leopard that prevents this.
Perhaps you have iPhoto '08 and are having one of the following problems? Both links lead to a problem description and potential fix:
I suggest you try Picasa
What version of iPhoto do you have presently installed?
iPhoto is part of iLife and if you have say iPhoto '08 (version 7.x) you can't only update iPhoto to the 2009 (version 8.x) but you have to update the entire suite. The screenshot you posted is the iPhoto Updater for iPhoto 8.1 so it requires iPhoto 8.1 to be installed.
The iLife '09 installer allows you to choose what applications you install. (Just click Customize on the Installation Type Screen)
The limitation is not a limitation of OS X 10.6.2+.
