Obviously dogmeat, who's apparently lived for over 100 years and travelled across the continent. Obviously his overly long life has taken a toll on his health and usefulness.
A cult worshipping the unexploded atom bomb. This was first in fallout 1, and the end result was (shock) you BLOW IT UP. This reappeared in Fallout Tactics, and hey ho it's back.
The Forced Evolutionary Virus to make supermutants. It's back, and it was apparently developed in a COMPLETELY DIFFERENT LOCATION (a vault, rather than the VATS (lol) from Fallout 1. It's also been made totally independantly and without communication, and yet has the same name, and same results.
BOTH main quests.
1) The purifier quest. Fix the purifierwater chip. Also note there's logs saying the vault's waterchip is dying as well.
2) The G.E.C.K. quest. I don't even need to explain why this is a copy of fallout 2. Note the circumstances of 'gaining' a GECK are exactly the same, as is the MAJOR Deus Ex Machina that extends the plotline, with ENCLAVE intervention, immediately afterwards.
3) The rescuing your dad thing. This is somewhat new (there's similarities to Arroyo, but it's different as well). It's also the shittiest part of the main quest...hrm...
The 'return to the vault' quest. Congratulations! You've saved your vault! They are going to prosper now, and everyone is happy again. What's your thanks? Fuck off and never come back. Here's guessing Fallout 4 starts you in a tribal village. Actually, ignore that. There isn't any tribals whatsoever in glorious motherfucking america these days.
The ENCLAVE. Apparently you wiped out the president--and their main base--in Fallout 2. Apparently you were wrong. Furthermore being run by a 'president' TWICE makes no sense, as the computer has somehow (rather conveniently) gained the persona of a president by it's own will.
Getting captured and breaking out of the ENCLAVE base - how can you not liken this to infiltrating and breaking out of the base in Fallout 2.
Furthermore in the ENCLAVE base, finding a caged Deathclaw and breaking it loose, setting it upon the ENCLAVE.
The 'Vault Dweller'. Yep, head off to little lamplight and now you're being called the Vault Dweller.