Improve and better are more general words and can apply to a variety of situations. They can be used interchangeably in certain situations – to improve (or better) one’s rank.
Better is usually used to suggest only a slight increase.
Ex – His new job is better than the last one.
While improve suggests a greater increase.
Ex – His vocabulary has improved in the past few years.
Ameliorate and meliorate are formal words. They are applied more to conditions rather than to specific things. Both these words are exact synonyms and generally refer to conditions that are in need of correction.
Meliorate is an extremely literary word and it is not used often nowadays.
Ex – Her kind glance was able to somehow meliorate his pain.