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I have always used \geq, but recently saw a document where the person uses \ge.

Once compiled, they look the same, but I searched to see if there was any difference between the two or a guideline as to which one to use, and most places say they are the same, but this source says

For greater than or equal to use the \ge command. The \geq command means greater than or equivalent to.

which makes no sense to me.

The question is then: are there any differences between the two, or any recommendation as to which one to use (what I have in mind is, for example, the case of eqnarray vs align, or $$ vs \[ \], where only the second option in each case should be used).

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No. That quote is not correct.

In Plain TeX, \le is let to \leq and \ge is let to \geq, so they are equivalent.

The LaTeX kernel contains the following lines:

\DeclareMathSymbol{\leq}{\mathrel}{symbols}{"14}
\let\le=\leq
\DeclareMathSymbol{\geq}{\mathrel}{symbols}{"15}
\let\ge=\geq

so the pairs \le,\leq, and \ge, \geq are equivalent in standard LaTeX.

As you can see, \ge and \geq are just different names for the same object (and similarly for \le and \leq). Both \ge and \geq stand for "greater than or equal to" and both \le and \leq stand for "less than or equal to".

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    Which means they look the same visually, but do they mean to represent different mathematical concepts as suggested by the quote above? Would you know?
    – Vivi
    Mar 26, 2012 at 0:42
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    No, they are exactly the same and both represent the same concept; \ge and \geq are different names for the same object (and similarly for \le and \leq). Both \ge and \geq stand for "greater than or equal to" and both \le and \leq stand for "less than or equal to".
    – Gonzalo Medina
    Mar 26, 2012 at 0:47
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    @Vivi: A remark about the message you quoted in the question: if you look at the next message of that thread (nfbnet.org/pipermail/blindmath_nfbnet.org/2008-May/001175.html) the author of the message you quoted recognizes that he made a mistake and that both names represent the same object.
    – Gonzalo Medina
    Mar 26, 2012 at 0:52
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    @Vivi: you can find the commented LaTeX kernel in the source2e.pdf file. In your system: texdoc source2e or in CTAN: ctan.org/pkg/source2e
    – Gonzalo Medina
    Mar 26, 2012 at 0:54
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    @Vivi There are two names because Knuth prefers \le and \ge and \ne, but realized that people may prefer \leq, \geq and \neq for semantical reasons. So he provided both variants in Plain TeX and LaTeX followed suit.
    – egreg
    Mar 26, 2012 at 5:45

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