Let's look at what 争气 and 出息 mean
①Refers to a dying man refuse to give up breathing ，②Achive success when being counted out
In other words, "show one's own worth/ make oneself proud"
出息: 前景或志向。Prospects or ambitions
不争气 --> not show one's own worth/ not make oneself proud --> underachieving
没出息 - A person who is deemed mentally weak, physically useless, can't do anything right or chose the right path, and has no chance/willingness to be excel. (一般用於責備男子和男孩. 被指責的人多被視為懦弱無能, 或不學無術, 不具備致富達貴的條件).
不争气 - A (young) person whose performance is not up to the expectation of being the lead or ahead of his peers. (中國人好面子. 兒女不高人一等就覺得沒面子, 此等人多會責怪子女不能替父母爭口氣. ...
Very similar, but 没出息 is way more critical than 不争气.
没出息 is a state/characteristic judging, the kid is worthless in the past, now and probably in the future.
不争气 is usually used under a specific failure, judging the performance or result.
"我真不争气": I've disappointed you.
"我真没出息": I am a good-for-nothing.
Very similar. If I were to translate,
This kid will never amount to anything.
This kid will never do me (or himself/herself) proud.
不争气 carries a hint of affecting others and/or comparing with others whereas 没出息 is just about the kid.