Why doesn't the code produce the desired result?
The font in the output is a so-called proportional font, which means that letters do not all have the same width, instead the only occupy the space they need. An 'i' takes up less width than an 'X' or an 'M' in fonts like that.
You can already see the effect of that in the first three lines: 'Tomas', 'Maine' and 'Patty' all have the same number of letters, yet they do not not line up completely at the end. That's because the letters have different widths, 'Patty' has two smaller 't's and 'Maine' a thin 'i', but the letters in 'Tomas' are all fairly wide.
'Springfield' is a very good example: It contains the small letters 'i' and 'l' and additionally a 'fi' ligature that makes the already fairly slim 'f' and 'i' come even closer together. In the output
MMMMMMMMMMM
would be considerably wider than
Springfield
with the same number of letters.
So you can't align the words perfectly just by counting characters and spaces.
If you look at the code
\begin{enumerate}
\item Tomas\verb+ +Thomas
\item Maine\verb+ +Main
\item Patty\verb+ +Patti
\item Roberts\verb+ +Robertson
\item Springfield\verb+ +Springvale
\item Nixon\verb+ +Dixon
\end{enumerate}
the words line up precisely because the code is shown in a different, monospaced font where all characters have the same width.
If you tell LaTeX to use a monospaced font with \ttfamily, stuff lines up
\documentclass{article}
\usepackage[T1]{fontenc}
\begin{document}
\begin{enumerate}\ttfamily
\item Tomas\verb+ +Thomas
\item Maine\verb+ +Main
\item Patty\verb+ +Patti
\item Roberts\verb+ +Robertson
\item Springfield\verb+ +Springvale
\item Nixon\verb+ +Dixon
\item MMMMMMMMMMM
\end{enumerate}
\end{document}
because all characters and spaces (\verb switches the font to the same font as \ttfamily) have the same width.

Why is \verb the wrong tool here?
As the example shows, \verb is the wrong tool here, because it would only align the words with a number of literal spaces instead of aligning the words via a more robust method as discussed in the great answers you got in How to align words on one line to the words on the next without resorting to table.
It's the equivalent of holding the space bar in MS Word until the words roughly line up instead of using TAB or some tabular method.
The main use case for \verb is to typeset text (usually computer code) containing special characters that normally needs escaping in TeX (\, #, _, ^, &, % etc.). See Escape character in LaTeX. It was not intended to be used as a way to align arbitrary text surrounding it.
\ttfamilyafter\begin{document}– Steven B. Segletes May 01 '20 at 10:13\ttfamilybefore theenumerateyou get perfect alignment because all letters have the same width. – moewe May 01 '20 at 10:14\verbis not supposed to be used for spacing. – egreg May 01 '20 at 10:23\verbin the mix, you'll find that the first three words have the same number of letters and that in the code as shown on this site they occupy the same width (because code on this site uses a monospaced font), but in the output the three words don't have the same width, because your output uses a proportional font. Just to be clear: the primary use of\verbis not to stop LaTeX from collapsing space tokens, it's intended to let you output exactly what you type without having to escape special characters. It's mainly used for writing about (LaTeX) code. – moewe May 01 '20 at 10:24