CSS Flexbox vs Box Model – Key Differences Explained

These are different styles.
display: box; is a version of 2009.
display: flexbox; is a version of 2011.
display: flex; is the actual version.

Quote of Paul Irish

Warning: Flexbox has undergone some major revisions, so this article is out of date. In summary, the old standard (Flexbox 2009), which this article is based on, was implemented in Chrome since v4, Firefox since v2, and IE10 beta.

Since then, the new flexbox standard debuted and started to debut in Chrome 17. Stephan Hay has written a guide on the new flexbox implementation. Since then, the spec underwent naming changes which started landing in Chrome 21. Chrome 22 has a stable implementation of the latest spec.

At this point, implementing either has its risks, so be aware.

Here is the blog about the different flexbox statements.
This is a blog entry by css-tricks about the different versions.

When i use flexbox, i always write it like that:

display: -webkit-box;
display: -moz-box;
display: -ms-flexbox;
display: -webkit-flex;
display: flex;

Edit:
Still not everyone has a browser/device capable of viewing flexbox layouts. So for fallback solutions or alternatives there is this article by Kenan Yusuf on how to use flexbox without using flexbox.

display: inline-flex does not make flex items display inline. It makes the flex container display inline. That is the only difference between display: inline-flex and display: flex. A similar comparison can be made between display: inline-block and display: block, and pretty much any other display type that has an inline counterpart.¹

There is absolutely no difference in the effect on flex items; flex layout is identical whether the flex container is block-level or inline-level. In particular, the flex items themselves always behave like block-level boxes (although they do have some properties of inline-blocks). You cannot display flex items inline; otherwise you don't actually have a flex layout.

It is not clear what exactly you mean by "vertically align" or why exactly you want to display the contents inline, but I suspect that flexbox is not the right tool for whatever you are trying to accomplish. Chances are what you're looking for is just plain old inline layout (display: inline and/or display: inline-block), for which flexbox is not a replacement; flexbox is not the universal layout solution that everyone claims it is (I'm stating this because the misconception is probably why you're considering flexbox in the first place).

¹ _{The differences between block layout and inline layout are outside the scope of this question, but the one that stands out the most is auto width: block-level boxes stretch horizontally to fill their containing block, whereas inline-level boxes shrink to fit their contents. In fact, it is for this reason alone you will almost never use display: inline-flex unless you have a very good reason to display your flex container inline.}