There is no real difference (both are based on the same underlying function), but as noted in the comments, they have some different default values (index_col is 0 or None, parse_dates is True or False for read_csv and DataFrame.from_csv respectively) and read_csv supports more arguments (in from_csv they are just not passed through).
Apart from that, it is recommended to use pd.read_csv.
DataFrame.from_csv exists merely for historical reasons and to keep backwards compatibility (plans are to deprecate it, see here), but all new features are only added to read_csv (as you can see in the much longer list of keyword arguments). Actually, this should be made more clear in the docs.
from_csvwill default to use the first column as an index.read_csvdefaults toNoneso an index will be created.read_csvshould be best suited for most datasets.from_csvexisted, personally having looked at both I would useread_csvas it has far more options that should assist with data manglingpandas.DataFrame.from_csvhas now been removed. I'm unable to call it and your link gives a 404: Not Found error.pandas.DataFrame.from_csvhas been deprecated since version0.21.0.