If I contribute $100 to my 403 B at work and decrease my withholding by $100 on the W4, will my tax bill come April be the same?

No, that is not a one-to-one change.  

Contributing to the 403B at work will reduce your taxable income, but it will not reduce your taxes by the amount that is contributed.

For example, suppose your income tax rate is 10%.  Then reducing your taxable income by $100 would reduce your taxes by $10 (10% of the $100).  This means that if you wanted to make a one-to-one change in your W4, then you could only reduce it by $10, not the $100 that was contributed to the retirement plan.