What is the difference between "to allege" and "to claim"?

Alleges means:

to state something as a fact but without giving proof

It is very similar to claims, but it is used in situations where there is a legal accusation which has not yet been resolved. For example, Apple here alleges Samsung copied technology. This may or may not be true, but in any case it is a legal claim.

You could use claims, but it wouldn't have the same connotations.

Merriam-Webster's Dictionary of Synonyms 1984) puts allege in a group of verbs with adduce, advance, and cite; and it puts claim in a group of verbs with demand, require, and exact. Here are its treatments of the two verbs:

adduce, advance, allege, cite may be used interchangeably in the meaning to bring forward by way of explanation, proof, illustration, or demonstration; however, they usually are clearly distinguishable in their implications and in their idiomatic associations. ... Allege may indicate a bringing forward or stating as if needing no proof {younger scholars nevertheless can allege a very strong point on their side—H. M. Jones} It may on the other hand stress doubt about or a disclaimer of responsibility for the truth of matter under discussion {those whose senses are alleged to be subject to supernatural impressions—Le Fanu} Its participial adjective alleged, especially, often serves as disclaimer of responsibility for the assertion {an alleged miracle} {the alleged thief} {the presence, real or alleged, of some hostile group—Dewey} ...

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Demand, claim, require, exact are comparable not as close synonyms but as sharing the basic meaning to ask or call for something as due or as necessary or as strongly desired. ... Claim implies a demanding either of the delivery or concession of something due one as one's own, one's right, or one's prerogative or of the admission or recognition of something which one asserts or affirms; thus, one who claims a piece of property demands its delivery to him as his own; one who claims that he has solved a problem demands recognition of the truth of his assertion {there is no right to freedom or life. But each man does claim such freedom—Alexander} {a genius, say his detractors, can be perverse, and they claim the right to tell this genius when and where and why he is perverse—Read}

Harry Shaw, Dictionary of Problem Words and Expressions 1975) likewise puts alleged and claim in different word groups: alleged with accused and suspected, and claim with demand. Here are the discussions of the relevant words in this dictionary:

alleged, accused, suspected. Each of these words is used to refer to persons involved in legal problems. To allege means "to state positively," "to assert without proof," or "to urge as an excuse or reason." ... One cannot allege a swindler, but rather a condition or a crime. ...

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demand, claim. ... Claim means "to assert a right": "The driver claimed that he was entitled to a hearing." Claim should not be used when you mean say, assert, state, or declare unless a right is involved: "The student claimed his right to an examination." "The student asserted (or said or stated or declared) that he was going to college."

And finally S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968) places allege with affirm, assert, asseverate, aver, avouch, avow, maintain, and testify and places claim with demand, ask, exact, order, request, and require. Here are Hayakawa's entries for the words:

assert, affirm, allege, asseverate, aver, avouch, avow, maintain, testify These verbs all mean to state positively, as though anticipating or countering argument or skepticism. Whereas assert means to state with some force or conviction, allege means to state without offering proof: It was alleged that he was present at the scene of the crime, but he asserted that he was in Europe at the time. Whatever one asserts he would defend in argument, but whether anyone believes something alleged is beside the point; the aim of alleging is to learn the truth by proving or disproving the claim made. ...

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demand, ask, claim, exact, order, request, require These verbs all refer to rather forceful communications between a speaker and another person. ... Claim and exact involve a slightly different situation than these other words. Claim suggests that a right has been asserted: claiming this land in the name of the queen; unions that claimed a right to share in the company's profits.

It thus appears that all three of these reference works treat allege as fundamentally involving an assertion of factuality without proof, and claim as fundamentally involving an assertion of a right, with or without supporting evidence.

In real-world practice, of course, most people make no such distinction between the two words. Here are their entries in Merriam-Webster's Eleventh Collegiate Dictionary (2003):

allege vt (14c) 1 archaic : to adduce or bring forward as a source or authority 2 : to assert without proof or before proving {the newspaper alleges the mayor's guilt} 3 : to bring forward as a reason or excuse

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claim vt (14c) 1 a : to ask for esp. as a right {claimed the inheritance} b : to call for : REQUIRE {this matter claims our attention} c : TAKE [in the sense of "to put an end to (life)" or "to remove by death"] {the accident claimed her life} 2 : to take as the rightful owner {went to claim their bags at the station} 3 a : to assert in the face of possible contradiction : MAINTAIN {claimed that he'd been cheated} b : to claim to have {organization ... which claims 11,000 ... members —Rolling Stone} c : to assert to be rightfully one's own {claimed responsibility for the attack}

Some of the two words' meanings are obviously nonoverlapping. But "to assert without proof or before proving," "to bring forward as a reason or excuse," and "to assert in the face of possible contradiction : MAINTAIN" seem to me to have so much in common that it's really difficult to tell which of these definitions go with allege and which with claim. The relative newcomer among these three definitions in the one for claim—"to assert in the face of possible contradiction : MAINTAIN"—which the Seventh Collegiate (1963) lists in identical language, but which the Sixth Collegiate (1949) omits, as you can see here:

claim v.t. 1. To ask for, or seek to obtain, by virtue of authority, right, or supposed right; to demand as due. 2. To assert as a fact, right, or relation which ought to to be acknowledged; as to claim the championship. 3. To call for; require; as, the subject claims our attention.

The Sixth Collegiate's definition 1 of claim clearly corresponds to the Seventh's (and Eleventh's) definition 1(a); the Sixth's definition 2 to the Seventh's and Eleventh's definition 2, and Sixth's definition 3 to the Seventh (and Eleventh's) definition 1(b). That leaves definition 3 of the Seventh (and Eleventh) with no counterpart in previous editions.

I have no doubt that English speakers and writers were using claim in the sense of "to assert in the face of possible contradiction" long before 1963, but they did so against the advice of usage commentators who argued that proper use of claim as a verb always implied some sort of demand for vindication of an underlying right or prerogative. Because English speakers and writers often chose to use claim in a sense extremely similar to "to assert without proof or before proving," they brought the word into close orbit with allege, leading to questions such as the OP's here.

To allege is always to accuse a party (human or corporate) of wrongdoing. You may, however, claim something without the accusation of wrongdoing.

For example:

• You can claim that a new product is far superior to its competitors
• You can claim that a storm damaged your conservatory by blowing your neighbour's tree down so that it fell on the conservatory
• You can claim that the dog ate your homework

In none of these situations are you accusing anyone of anything (except, possibly, the dog. But he isn't a legally responsible entity and you aren't asking your teacher to take action against him).

You can claim that someone is guilty of wrongdoing against you but this can be less aggressive that accusing them of it. For example:

• You can claim that a rival's product breaches a patent you hold in some significant detail.
• You can claim that a neighbour's tree is damaging your conservatory because the branches are scratching the glass and wearing away the frame
• You can claim that your sister spilt coffee over your homework

If you claim negligence or deliberate wrongdoing on the part of another this is the same as alleging it:

• You can claim or allege that a rival has copied your patented product deliberately.
• You can claim or allege that your neighbour has cut down his tree so carelessly or incompetently that it has fallen on your conservatory
• You can claim or allege that the class bully has thrown your homework into the river.

There is an overlap in the meaning but allege is stronger, more associated with criminality and tends to imply a higher standard of proof. Claim is wider in application, slightly gentler and somewhat less formal in its implications.