Yhyjyk

What is the difference between command a[bc]d and command ab,cd? Why do people use ab,cd when there is already a[bc]d?

a[bc]d is a filename pattern. It will expand to abd and acd if those are names of existing files in the current directory. If abd exists, but acd does not, then it would only expand to abd, and vice versa. By default, the pattern would remain unexpanded if there was no matching filename.

ab,cd is a brace expansion (in shells that support these). It will expand to the two strings abd and acd. This is regardless of whether these strings corresponds to existing filenames or not.

If you are constructing strings, use a brace expansion. If you are matching filenames, use a filename pattern.

a[bc]d is pattern-matching, and is part of the POSIX standard. In POSIX, this is introduced as the "pattern bracket expression". It is documented in section 2.13 of the manual

When unquoted and outside a bracket expression, the following three
characters shall have special meaning in the specification of
patterns:

?

A question-mark is a pattern that shall match any character.

*

An asterisk is a pattern that shall match multiple characters, as
described in Patterns Matching Multiple Characters.

[

The open bracket shall introduce a pattern bracket expression.

Section 2.13.3 also mentions something that it behaves differently from what one would expect for usual regexs when it is used for filename expansion (emphasis by me)

The rules described so far in Patterns Matching a Single Character and
Patterns Matching Multiple Characters are qualified by the following
rules that apply when pattern matching notation is used for filename
expansion:

The slash character in a pathname shall be explicitly matched by using
one or more slashes in the pattern; it shall neither be matched by the
asterisk or question-mark special characters nor by a bracket
expression. Slashes in the pattern shall be identified before bracket
expressions; thus, a slash cannot be included in a pattern bracket
expression used for filename expansion. If a slash character is found
following an unescaped open square bracket character before a
corresponding closing square bracket is found, the open bracket shall
be treated as an ordinary character. For example, the pattern
"a[b/c]d" does not match such pathnames as abd or a/d. It only matches
a pathname of literally a[b/c]d.

ab,cd is braces expansion, it is not in the specification by POSIX. Here is the corresponding part from the bash manual (emphasis by me):

Brace expansion is a mechanism by which arbitrary strings may be
generated. This mechanism is similar to filename expansion (see
Filename Expansion), but the filenames generated need not exist.
Patterns to be brace expanded take the form of an optional preamble,
followed by either a series of comma-separated strings or a sequence
expression between a pair of braces, followed by an optional
postscript. The preamble is prefixed to each string contained within
the braces, and the postscript is then appended to each resulting
string, expanding left to right.

According to the comment by @mosvy, this first appeared from csh but the behavior in bash is different from csh and other shells. This type of braces expansion is also present in glob(3).

There is another type of braces expansion a..z that only appeared after bash 3.0, and there are more added in bash 4.0.

In a shell where globbing is turned on, execute in a empty folder, the following result is returned

$ echo a[bc]d
a[bc]d
$ echo ab,cd
abd acd

In response to @Jesse_b's comment, if you are in an interactive shell and both of them applies, a[bc]d is less trouble typing. For example grep pattern [ab][12].txt.