What is the difference between over and frac?

Question

What is the difference between
{some-expression}over{another-expression}
and
frac{some-expression}{another-expression}
?

Which is preferable?

Mico · Accepted Answer

The command over is a so-called "primitive" command that's available in Plain TeX and in LaTeX. frac is a LaTeX-only command that builds on the over macro to provide something that's much less likely to throw unexpected errors if the user doesn't handle the syntax of the over command just right.
Assuming you have the amsmath package loaded, the frac command is defined as follows:
DeclareRobustCommand{frac}[2]{{begingroup#1endgroup@@over#2}}

where @@over is an amsmath-internal version of over. Note that the first argument of frac (the numerator) is carefully encased in a begingroup ... endgroup construct and that the entire command's "content" is surrounded by a double set of curly braces. The braces serve to remove potential ambiguities that might arise from a less than fully careful use of the over command by itself.
If you're a user of LaTeX, there's no reason whatsoever for not using the frac command -- and several good reasons for not using the over command directly. In short, if you use LaTeX, use frac.

egreg · Answer

The latter is preferable. It corrects a behavior that in some situations can reveal itself in a bad way. The LaTeX definition is

deffrac#1#2{{begingroup #1endgroupover #2}}

In some cases an assignment made in the numerator can affect also the denominator. It's difficult to produce it with standard LaTeX commands, but you can compare the result of

[
fam0 aover b
]

with

[
frac{fam0 a}{b}
]

In the first case both "a" and "b" will be upright, which is probably not expected.

Another relevant argument against the over syntax is that it's foreign to LaTeX, which always uses first the command and then its argument and never an "infix" syntax. Moreover the grouping around numeratorover denominator is automatically provided, avoiding mistakes such as

1over 2 over 3

Update

Starting from the 2019-10-01 release of LaTeX, the definition of frac uses DeclareRobustCommand, as part of a program to remove as many fragile commands as possible:

% latex.ltx, line 4705:
DeclareRobustCommandfrac[2]{{begingroup#1endgroupover#2}}

In olden times it was not possible to make all commands robust, because of memory restrictions. These considerations are no longer a problem.

The amsmath package has used DeclareRobustCommand for its modified definition of frac for many years.

Américo Tavares · Answer

In User’s Guide for the amsmath Package p.14 it is written the following:

The primitive generalized fraction commands over,
  overwithdelims, atop, atopwithdelims, above,
  abovewithdelims produce warning messages if used with the amsmath
  package, for reasons discussed in technote.tex.

And in Technical notes on the amsmath package p.2:

Not only is the unusual syntax of the TeX primitives rather out of
  place in LaTeX, but furthermore that syntax seems to be responsible
  for one of the most significant flaws in TeX's mathematical
  typesetting capabilities: the fact that the current mathstyle at any
  given point in a math formula cannot be determined until the end of
  the formula, because of the possibility that a following generalized
  fraction command will change the mathstyle of the preceding material.
  
  (...) 
  
  There are additional bad consequences following from the syntax
  of those generalized fraction commands that only become evident when
  you do some writing of nontrivial macros for math use. For example, as
  things currently stand you cannot measure the size of any object in
  math without going through mathchoice and leaving and reentering math
  mode via hbox{$ (which then introduces complications regarding
  everymath and mathsurround). And it seems that uncertainty about the
  current mathstyle is the only barrier to allowing the use of mu units
  with vrule, to make vertical struts in constructing compound symbols
  or notation. And so on and so forth.

What is the difference between over and frac?

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