Jonah: The two you’ve mentioned are the two BIG differences:
(1) Then–immigration from afar. Now–Immigration from next door.
(2) Then–no welfare state. Now—USA=FreeLunchistan.
The history of immigration into the USA can be briefly encapsulated thus: 1776-1846–very little. 1846-1924–First Great Wave. 1924-1965–very little. 1965 on–Second Great Wave. Only in that post-1965 wave has the welfare state been a factor.
Some of the lesser differences (legal/illegal, work-seeking/family-unification) are important, but those two are the biggies.