I'm looking for a more succinct word/small phrase that conveys the idea of "comparing and contrasting". Is there a word that hits two birds with one stone by representing both similarities and differences between two ideas/objects?
The Collins Dictionary & Thesaurus (2000) provides seven meanings of 'compare', three of which involve '[examining] in order to observe resemblances and "differences"', and weighing something against something else.
The other meanings of 'compare' are generally used to cast the person or item under comparison in a positive light e.g. 'the general has been compared to Napoleon'. Thus used in most senses 'compare' means more or less the same thing as 'contrast', which Collins specifies as one of its synonyms.
Under 'contrast' Collins refers you back heavily to 'compare' both in its meanings and its synonyms.
From all that, I deduce that it is difficult to 'contrast' without 'comparing', but that it is possible to use the word 'compare' without it involving any specific 'contrasting'. e.g. Using the word as an abstract noun one can say: 'It is a motor car beyond compare.'
The two words thus stand in a very close relationship to one another and in a large number of cases are interchangeable. But I can see why students of English might be led to believe there is a significant difference. So many times the expression 'compare and contrast' is used as though the words said different things. Among the biggest culprits are examiners who put questions like 'Compare and contrast the roles of nationalism and socialism in European history since 1900'. They make it sound as though there was a greater difference between compare and contrast than actually exists.
The two words mean virtually the same thing.