Welcome to the board.
The best way that I know of to do this - in fact, the only way that I know how to do it - involves a few easy-to-follow steps.
1. Start from the worksheet that you have decided is going to be your "Master" sheet. Make a copy of it so that your later data changes don't mess up anything.
2. In the C or D column of your Master worksheet, perform the VLOOKUP of the A value on that sheet into the second sheet to get the applicable date from the second sheet (if a date exists, and if the A value also exists on that sheet).
3. After your VLOOKUP is done and you now have three columns of data: the Items in A and their dates (the original two-column Master sheet) plus a column of dates from the VLOOKUP.
4. Write a formula in another column to the right to compare your "Master" dates against your VLOOKUP dates and resolve "which one wins" based on your rules. (Be sure to consider that there may be Items on your Master list that don't exist on your secondary sheet - and vice versa - and be sure to consider that some dates may have been entered incorrectly on either list - as text, for example, and not numbers in date format - or dates may be missing on one list or the other, etc. In other words, examine your data to check for various error conditions.)
5. When you are sure that you have a good list of Master items (Column A) and "winning" date values to apply in the far right column of your worksheet, copy those winning dates and do a "Paste Special / VALUES" onto your original Column B, check the results, wipe out your added columns, and you're done.