Hi bryanktracy1!
The new withholding tables and method of calculating the withholding can be tricky for employers to learn.  It is quite possible that the new employer had the withholding calculating incorrectly.  The Form W-4 and method of calculating withholding changed in January of 2020.  Employers are still learning the correct calculations.  If all the information was virtually the same (salary and W-4 selections), then both employers should calculate roughly the same amount of tax.  I'm glad that your wife was able to submit a new W-4 with the "additional" amount so that she can have some withholding coming out of her paycheck.  You may be able to find a paycheck calculator tool online to check the withholding.  Here's a link to one that you may want to try:

https://www.adp.com/resources/tools/calculators/salary-paycheck-calculator.aspx

There can be several reasons why no federal income tax was withheld from an employees payroll.

The employer may have made a mistake when entering the W-4 withholding information into their payroll system. Or the employee may have indicated they are exempt from federal tax withholding (in error)  when they  completed the Form W-4. You can request a copy of the W-4 from your employer to see if either of either of these scenarious occurred.

In some cases, however, an employee may actually be exempt from withholding.  If an employee qualifies, he or she can also use Form W-4 to tell you not to deduct any federal income tax from his or her wages. To qualify for this exempt status, the employee must have had no tax liability for the previous year and must expect to have no tax liability for the current year.