I read that using HTML coding where CSS coding can be used is tantamount to feeding search engines deprecated attributes. If you know about what I am talking, then do you think that using HTML instead of CSS adversely affects a website's search engine ranking?

I reckon this article pretty tells you everything you need to know and which is best

I read the articles associated with the link. As long as the website is not penalized, much depends on one's personal preference.

You don't use HTML instead of CSS. Let me elaborate...

HTML is a markup language, by definition. HTML allows you to change the way data appears to the person viewing it.

CSS is intended to actually produce layouts and all visible characteristics of a web page. CSS layers should be used instead of HTML tables to produce a general website layout. HTML tables, though commonly used as layout mechanisms, were only intended for the organization of tabular data.

As far as "feeding search engines deprecated attributes".. search engine spiders strip HTML coding anyway. They index the "meat" of your pages, and they also parse the META information contained in many web page headers (slightly deprecated at the moment).