Unknown origin of additional spaces in a glyph

Arcs create wider bounding boxes, because they include the control points. You can reset the bounding box in various ways, here I present one.

See https://tex.stackexchange.com/search?q=tikz+controls+bounding+box for other examples. I chose the method in Specific drawings with arrows and vertices in latex by Henri Menke.

\documentclass{article}

\usepackage{tikz}
\usepackage{braids}

\usetikzlibrary{knots}

\newcommand{\firstcorrect}{%
  \begin{tikzpicture}[x=0.4cm,y=0.4cm]
  \path[use as bounding box] (0,0) rectangle (1,0.7);
  \draw[thin] (0,0) .. controls +(3,1) and +(-3,1) .. (1,0);
  \end{tikzpicture}%
}
\newcommand{\secondcorrect}{%
  \begin{tikzpicture}[x=0.4cm,y=0.4cm]
  \path[use as bounding box] (0,0) rectangle (1,0.7);
  \begin{knot}
  \strand[thin] (0,0) .. controls +(3,1) and +(-3,1) .. (1,0);
  \end{knot}
  \end{tikzpicture}%
}

\begin{document}

Braids diagram with \texttt{tikz} library \texttt{knots}: 
\secondcorrect; with box: \fbox{\secondcorrect}

Braids diagram in plain \texttt{tikz}: \firstcorrect;
with box: \fbox{\firstcorrect}

\end{document}

Note that you were including several spaces by not protecting the end of lines.

It's also not appropriate to draw the picture with a thick pen and then scale it: you get better control without scaling.

Maybe you want to set the bounding box slightly wider.

Maybe not the best, but I propose some clipping of the second image; I think that the control points (or something like that) are far away:

\documentclass{article}
\usepackage{tikz}
\usepackage{braids}

\usetikzlibrary{intersections}
\usetikzlibrary{knots}

\newcommand{\correct}[1]{
\begin{tikzpicture}[#1]
\draw[xshift=5cm] (0,1) .. controls +(0,0) and +(0,0) .. (0,1) .. controls +(1,0) and +(-1,0) .. (2,0);
\draw[xshift=5cm] (0,0) .. controls +(0,0) and +(0,0) .. (0,0) .. controls +(1,0) and +(-1,0) .. (2,1);
\end{tikzpicture}
}
\newcommand{\notcorrect}[1]{
\begin{tikzpicture}[#1]
\clip (-3ex,-3ex) rectangle (4em,7ex);
\draw(0,0) .. controls +(3,1) and +(-3,1) .. (1,0);
\end{tikzpicture}
}

\begin{document}

This is a correct \correct{scale=0.3} input. This is not a correct \notcorrect{scale=0.3} input.

\end{document}

Result: