The answer to this question depends on a number of things:
- How backwards compatible do you need to be? Including IE6 will decrease the capacity of pure CSS; and
- How much of your site is fixed-width and/or fixed-height. There are certain things in CSS that become hard if not impossible in variable width and/or height situations.
Side-by-side content is a problem for CSS. You can use floats for this but if the sum of widths exceeds the width of the container, the tail end floats will fall down below. Tables are more capable in this regard as they will squeeze columns where possible to make things fit and cells will never be split onto new rows.
Vertical centering you mentioned. Its trivial with tables and hard or impossible (depending on compatibility and fixed or variable heights of the container and the item) in pure CSS.
You may also be referring to hover content. IE6 only supports the :hover pseudo element on anchors. Javascript is required for that browser for :hover-like behaviour.
Basically if what you need to do can be done fairly trivially with pure CSS then do it. If not, don't feel bad if you have to use tables despite all the anti-table fanatics (and they are fanatics) jumping up and down in horror.
If you want a relatively simple exmaple of this check out Can you do this HTML layout without using tables?. This is a conceptually simple layout problem that is trivial with tables and noone has yet posted a solution meeting the requirements with pure CSS.