I'd probably set a filter or sort to get the new items I need to
copy, then use datasheet view to copy the cells and paste them into
the same list on the new server when it is also in datasheet view.
The major thing I've found is that the fields have to be identical
or else you can screw up your records.
I did this yesterday, and I made sure both views were identical,
then switched to "edit in datasheet" and highlighted and copied the
cells I needed. I suppose you could also select the entire rows, but
I didn't do that. I then switched to the new list and clicked inside
the first cell of the last row (the empty one) and pasted. It
plopped everything into the proper place.
Formatting might be an issue for certain types of columns. Mine were
pretty simple ones, so I didn't run into any snags.
Quick note: Something I recently discovered is that you cannot edit
in datasheet (it becomes read-only) *if* you have content approval
required for the list. I needed to switch this back to off (after
making sure nothing was waiting for approval) so I could use the
edit in datasheet feature.