Out of sight, out of Ear, as the saying might go. Despite sharing a common language, American and British English sound distinctive from each other. To be fair, each of the regional variances in these countries also has a unique quality, but that does not change the truth. In terms of speech, over time, Yankees and Limies learn to understand each other quite well (maybe with the help of a local interpreter). In terms of writing, writers must take into account the differences between the US and UK in composing texts.
The first aspect to come to mind is spelling. The actual differences are minor but nevertheless noticeable. The American ize is written with an ise in England, as in rationalize/rationalise and capitalize/capitalise. The English add the letter u in the middle of the or combination, creating honour and flavour. Finally, Americans insist on double letters in certain words while the British are satisfied with only one, as in fulfill/fulfil and enroll/enroll. There are many others, such as center/centre, but these words remain highly recognizable to people from both sides of the Atlantic.
In terms of vocabulary, there is a long list of items that have been given different words in the two dominant forms of English. Some are specific to the country. For example, a bonnet, lorry, flat, and sweets in London are a car hood, truck, apartment, and dessert in Washington D.C. Specific fields, such as accounting, have quite a long list of differences. Trickier, some words used by both sides have different meanings. The classic example is when a Yorker eats chips and crisps, a New Yorker is consuming fries and chips. Slang is by definition local, even within a country.
For the writer, a more important difference is the accepted writing styles. American paragraph writing emphasizes a very strong topic sentence, the first sentence in a paragraph, declaring the topic and subtopics to be discussed, with a general summary sentence at the end. By contrast, British paragraphs often begin with a vague opening statement but end in a thorough restatement. Also, American usage allows a comma (,) before the words and and or in a series of items, but British usage does not require it as in blood, sweat and tears.
Writers need to be aware of these differences. However, in terms of actual difficulty, they are a drop in the bucket as compared to the overall challenge of writing good English even for a native speaker. So, as the French say, vive la difference!