Why different bra ket notations?

The macros with uppercase initial are "self-expanding" based on the contents, as the documentation says. Just like it's not good to always use \left and \right, it's also good to choose with care between \Bra and \bra.

As far as the additional spacing is concerned, a solution is to load mleftright:

\documentclass{article}
\usepackage{amsmath}
\usepackage{braket,mleftright}
\mleftright

\begin{document}

\begin{align*}
\Bra{\frac{1}{\sqrt{2}}\left(\uparrow + \downarrow\right)}
  &=\frac{1}{\sqrt{2}}\left(\Bra{\uparrow}+\Bra{\downarrow}\right)\\
\bra{\frac{1}{\sqrt{2}}\left(\uparrow + \downarrow\right)}
  &=\frac{1}{\sqrt{2}}\left(\bra{\uparrow}+\bra{\downarrow}\right)\\
&=2 222 22\\
&=2\mathinner{222}22\\
\end{align*}

\end{document}

Generically speaking the \left<delim> ... \right<delim> are used to fit the height of the content with expandable delimiters, chosen to by higher than all parts of the content.

But it is well-known that this choice often produce too large (tall) delimiters, and must be avoided for aesthetic reasons (and resulting typographic rules), as tactfully said by egreg and as exemplified by Aditya's comment. It's better practice to keep the delimiters height as small as possible, provided they ensure good readability.

That is why I don't use braket.sty but my own .sty based on \big and \Big, and some trick related to old-fashion \def with e.g. for the bras:

\def\bra<#1|{\langle #1\rvert}
\def\Bra<#1|{\big\langle #1\big\rvert}
\def\BRA<#1|{\Big\langle #1\Big\rvert}

and I would code the example discussed here:

\BRA<\frac{1}{\sqrt{2}}\left(\uparrow + \downarrow\right)|
    =\frac{1}{\sqrt{2}}\big(\bra<\uparrow|+\bra<\downarrow|\big)

the result of which, is IMO, better looking than the braket.sty result as coded by strpeter and posted by egreg.
As further comments:

• it would typeset the fraction in \textstyle to avoid to give him the emphasize
• the left hand side is not a really meaningful notation for this state