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I have two SQL files, one is old.sql and the other one is new.sql.

Suppose old.sql contains a table with three fields, Emp_Id, Name and Address and data stored in old.sql as follows:

Insert into table1 values (101 ,"a", "xyz");

Insert into table1 values (102 ,"b", "pqr");

Then I have changed "a" address "xyz" to "xyz123" and saved that data in the new.sql file.
Now the new.sql file contains data as follows:

Insert into table1 values (101 ,"a", "xyz123");

Insert into table1 values (102 ,"b", "pqr");

When I use the diff command like this:

diff old.sql new.sql

it gives differences line-wise but I want only updated data, like xyz123.

You might find wdiff useful for this type of comparison; it's a front-end to diff which produces word-by-word comparisons. With your example it produces by default

Insert into table1 values (101 ,"a", [-"xyz");-] {+"xyz123");+}

Insert into table1 values (102 ,"b", "pqr");

It can use terminal features to make the output more legible on a terminal (wdiff -t). It also has a -3 option which limits output to changed words only:

======================================================================

 [-"xyz");-] {+"xyz123");+}

======================================================================

If you don't have wdiff already installed, you need to install it. Run sudo apt-get install wdiff or sudo dnf install wdiff or sudo yum install wdiff or the command that is appropriate to your operating system.

By definition, diff is showing differences lines by lines (see diff manual page), it will therefore not show only the differing characters.
You can reduce the amount of difference by pre-processing the files, for exemple by inserting an end-of-line character after each semi-column:

sed -e 's/;/;'$'n/g' old.sql > old.patched

Then use diff on the two resulting files.

You can use:

diff -u old.sql new.sql |colordiff |diff-highlight

colordiff is a Ubuntu package. You can install it using sudo apt-get install colordiff.

diff-hight is from git (since version 2.9). It is located in /usr/share/doc/git/contrib/diff-highlight/diff-highlight. You can put it somewhere in your $PATH. Or get it from diff-so-fancy project.