First,
I noticed your column names had spaces in them. You should either remove those, or replace them with an _. Also, you should replace the "#" symbols with "num". Having only letters and numbers in a column and table name makes things a little less messy.
Try putting your Query string all on one line, and using the string joiner (.) to join all the strings.
I've found it beneficial to give your variables meaningful names. If it's a string put "str" or "string" in the variable name, for example, and what that string is for. More typing, I know, but it saves headaches.
Finally, if you have a string with a lot of escaped characters (namely quotes), use an ' to define the string. Makes everything look neater, and I believe it helps a bit with performance.
$personUpdateString = 'Update ' . $table_name . ' SET Name = "' . $Name . '", ExtNum = "' . $Ext . '", CellNum = "' . $Cell . '", DeptNum = "' . $Dept . '" WHERE Name = "' . $name . '";
Hope that helps.