Expropriate is generally used to imply removal by a heavy-handed, but legal force, often by government.
Appropriate merely means to take something over as one's own. Illegality is not usually implied, but it does carry a sense of firmly transferring something to oneself or another, or on the part of another person to themselves or another. But note from the examples how it is used to describe the actions of nature.
However misappropriate implies acquisition by illegal means - fraud or theft..
A few examples of the use of each of the three words, from the OED, should help explain.
1875 J. H. Bennet Winter & Spring Mediterranean (ed. 5) xiii. 480
The Government gives..a power to expropriate the owner of the land
1881 Macmillan's Mag. 44 132 To expropriate the owners from their
estates must be a very bitter pill.
1881 Daily Tel. 14 Feb. A corner of the garden..was ‘expropriated’
by Baron Haussman for the purpose of widening the Rue Lafayette.
1884 Contemp. Rev. Oct. 518 The State..expropriates private
property for public utility.
1876 E. Mellor Priesthood i. 15 The name ‘priest~hood’..was never
appropriated by apostles to themselves.
1785 W. Cowper Task v. 761 A liberty like his, who unimpeached Of
usurpation..Appropriates nature as his Father's work.
1871 J. Tyndall Fragm. Sci. II. vi. 83 The bud appropriates those
constituents..for which it has elective attraction.
1809 T. E. Tomlins Jacob's Law-dict. at Appropriation, The
monasteries..appropriated as many benefices as they could by any means
1801 J. Strutt Glig-gamena Angel-ðeod Introd. 7 These
amusements..were appropriated to the season of Lent.
1839 H. Hallam Introd. Lit. Europe III. vii. 634 The subject
chosen is appropriated to the characteristic peculiarities of the
1904 J. Conrad Nostromo ii. iii. 127, I am not likely to
misappropriate the funds.
1952 W. Plomer Museum Pieces (1961) xxvi. 204 He had
misappropriated the funds entrusted to him.
1987 Melody Maker 8 Aug. 10 Disfiguring and misappropriating pop's
1990 Times 2 Feb. 34/7 The purpose of the fraud was to enable
moneys to be ingathered and misappropriated.