Difference between the verbs "appropriate" and "expropriate"?

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.

Expropriate

1875 J. H. Bennet Winter & Spring Mediterranean (ed. 5) xiii. 480
The Government gives..a power to expropriate the owner of the land required.

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.

Appropriate

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 obtain.>

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 poet.

Misappropriate

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 legacies.

1990 Times 2 Feb. 34/7 The purpose of the fraud was to enable moneys to be ingathered and misappropriated.

Yes, as the grammarist notes, appropriate and expropriate are often used interchangeably in reference to goverment appropriation of private property:

To expropriate:

• is to deprive of possession, especially using eminent domain or judicial action. So expropriate is often interchangeable with appropriate where the latter involves depriving of possession, and both words are indeed commonly used in reference to government appropriation of private property and companies.

As in :

• The Columbia City Council yesterday approved a measure to appropriate funds that could be used to provide bonuses to city employees. [Columbia Daily Tribune]

To appropriate:

• is (1) to take possession of for one’s own use, and (2) to set something apart for a specific use. To appropriate something is not necessarily to deprive another of possession; for example, an artist might appropriate another’s style, or one might appropriate a catchy archaic phrase from the 18th century. But appropriation does sometimes involve depriving of possession, such as when a government appropriates a private company. In its second sense, appropriate usually applies to government funds or other official resources given out for specified purposes. This use of the word is primarily American, but it’s not unheard of elsewhere.