The two following pdfs are both typeset in LaTeX but clearly they differ in style. Default package I am using yields the same result as seen in second link. What should I do in order to get the style as in the first pdf?
Your assertion that the documents were set with latex is not correct, the first one is not set with latex but with amstex, which is a different, more or less obsolete, format.
amsart class (including
amsmath) is more or less a re-implementation of amstex for latex, so I would use that.