No they are not.
An exemption is part of the tax return. Each person has a personal exemption and dependents have a personal exemption. Each exemption excludes some income from taxation. And by the way, exemptions have been eliminated as part of the tax reform law starting this year (2018 tax year).
Allowances are used on the W-4 form to help you calculate withholding -- the concept is to figure out how much of your income will be covered by various deductions and credits, so you get the proper withholding.
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When there were still personal exemptions, an allowance was equal to an exemption -- each one removed $4000 from consideration as taxable income. But you also got exemptions for other things -- since the child tax credit reduces your tax, that was also worth an allowance, and you could get an allowance for every $4000 of itemized deductions you planned to claim. You also took away allowances if you had more than one job or your spouse worked, because of the way the tax tables make assumptions about taxable income.
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With the tax reform law, the IRS has not issued a revised W-4 yet, but they did change the withholding tables. Supposedly, if you fill out the W-4 worksheets for your situation, they will coordinate with the employer withholding tables so you get the correct amount of tax withheld.
