Erusin is a full Kinyan mid'Oraisa, l'Chumra and l'Kula. (If someone tries to be Mekadesh her after Eirusin from another man, she does not need a Get l'Chumra.)

Nisu'in defines the second part of marriag: making the wife part of one's household. This does not take place simply through Eirusin. Rather Eirusin "acquires" the woman and "reserves" her to eventually be brought into one's house. Until that point (i.e. Nisu'in) she gets to prepare for joining the husband's family usually for a 12 month period. (Nowadays, Nisu'in is done together with Eirusin.)

Nisu'in gives the husband priveleges of Yerushah and Hafaras Nedarim, and allows him to be Metamei to bury her if he's a Kohen. (It also entitles him to live with her like man and wife, as mentioned above.)

Be well,

Mordecai Kornfeld