The term co-belligerence usually indicates some kind of remotedness between the co-belligerent parties, cultural, ideological or otherwise, whereas alliance indicates a corresponding closeness.
Co-belligerence is also a less precise status than wartime partnership in a formal military alliance. A government "finds itself" in a position as co-belligerent compared to the alliance which is mostly actively and willfully sought.
The Allies as co-belligerents with former enemies
The term was used in 1943-45 during the latter stages of World War II to define the status of former German allies and associates (chiefly Italy, but also from 1944 Romania, Bulgaria, Hungary and Finland), after they joined the Allies' war against Germany.
Finland as co-belligerent with Nazi-Germany
Co-belligerence is of particular importance as the term used by Finland for her military co-operation with Nazi-Germany in the Continuation War of 1941-44, when both countries had the Soviet Union as a common enemy. The Continuation War was a direct consequence of Nazi-Germany's attack on the Soviet Union, Operation Barbarossa. Until then the German and Soviet governments had been allies, as during the Winter War (1940) against Finland.
While the Allies from 1941 often referred to Finland as one of the Axis Powers, this description is viewed as erroneous by some (especially in Finland), as Finland was never a signatory to the German-Italian-Japanese Tripartite Pact of September 1940.
Finland's co-belligerence as an euphemism
Hitler declared to be allied with the Finns, who in turn declared their intention to remain a neutral country. The truth was somewhere inbetween. In practice, the navy of "neutral" Finland contributed to Germany's attack from the beginning by mining the Gulf of Finland: