SHTML
There isn't any difference between using .shtml and .php files other than the languages in which they are parsed. .shtml contains SSI scripting, while .php contains PHP scripts. There isn't any specific reason to choose one over the other - it depends completely upon the developer.
DHTML
DHTML isn't a different language or script from HTML. It just refers to the combined use of a markup language (HTML), a client-side scripting language (JavaScript) and a styling language (CSS). It simply notifies that your HTML page has dynamic content.
None of these are deprecated. It is just that SHTML has gone out of use as many people have switched to PHP and other server-side languages, and that people have minimized the usage of the term DHTML for talking about dynamic HTML pages, as now most of the websites have dynamic content.