Difference between render :action and render :template

There is no difference.
render :template => 'some/thing' is the same as just render 'some/thing', as well as the same as render :action => 'thing' if we are in the some controller.

From Ruby On Rails guide;

render :edit
render :action => :edit
render 'edit'
render 'edit.html.erb'
render :action => 'edit'
render :action => 'edit.html.erb'
render 'books/edit'
render 'books/edit.html.erb'
render :template => 'books/edit'
render :template => 'books/edit.html.erb'
render '/path/to/rails/app/views/books/edit'
render '/path/to/rails/app/views/books/edit.html.erb'
render :file => '/path/to/rails/app/views/books/edit'
render :file => '/path/to/rails/app/views/books/edit.html.erb'