Federalists in New England wanted to secede from the Union and that's the reason they don't exist anymore. I'm no fan of either of the two 1800's parties, but the Democratic Party we know today was founded by Andrew Jackson and Martin Van Buren during the 1820's. The Democrats are liars when they say that they're the party of Thomas Jefferson.
They were almost all followers of Jefferson and/or old democrat/republicans. Very few of the latter shifted into the Federalists successor, the Whigs.
The same voters voted for the Democrats who voted for the Democrat/Republicans. Democrat power was concentrated in the slave power and urban north just as had that of the D/Rs.
There was no substantial difference other than dropping Republican from the name.
Martin Van Buren's base of power in New York was the same as that of George Clinton. Jackson explicitly campaigned upon Jeffersonian principles and proclaimed himself a Jeffersonian. The same class warfare ideology was employed with a vengence. Jackson was as opposed to the National Bank as was Jefferson demostrating the same economic blindness and fanaticism.