めくLike, To show signs of, To have the appearance of, To come to a (state) similar to, Somewhat, An air of, Feels almost like

Both describe something beginning to show a certain quality, but the nuance differs. めく focuses on a noticeable change in atmosphere or appearance, often used for seasons or general impressions, as in 秋めいてきた ('It has started to feel autumn-like.'). びる emphasizes the emergence of a specific visible or behavioral trait, often something concrete or character-related, as in 大人びた話し方をする ('He speaks in a grown-up way.').

Both describe something becoming like something else, but the nuance is different. めく・めいた means ‘showing signs of’ and refers to a noticeable change in atmosphere or appearance, without judging whether it is typical, as in 秋めいてきた ('It has started to feel autumn-like.'). らしい②, especially in らしくなってきた, focuses on becoming more typical or expected for that thing, emphasizing how well it matches the usual image, as in 秋らしくなってきた ('It has started to feel properly autumn-like.').