The decisive difference between tamannī and tarajjī is the expectation of attainment.
AspectTamannīTarajjī
ObjectA desired outcomeA hoped-for outcome
LikelihoodImpossible or remoteExpected / possible
Usual particleلَيْتَعَسَى، لَعَلَّ
Typical caseReturn of a past ageHope of coming good news
This is a guide to the intended function, not an inflexible vocabulary test. A particle usually indicating hope may be used figuratively for wishing.