I provided the following previously:
UTF
UTF is a family of multi-byte encoding schemes that can represent Unicode code points which can be reperesentative of up to 2^31 [roughly 2 billion] characters. UTF-8 is a flexible encoding system that uses between 1 and 4 bytes to represent the first 2^21 [roughly 2 million] code points.
Long story short: any character with a code point/ordinal representation below 127, aka 7-bit-safe ASCII is represented by the same 1-byte sequence as most other single-byte encodings. Any character with a code point above 127 is represented by a sequence of two or more bytes, with the particular of encoding best explained here.
ISO-8859
ISO-8859 is a family of single-byte encoding schemes used to represent alphabets that can be represented within the range of 127 to 255. These various alphabets are defined as "parts" in the format ISO-8859-n, the most familiar of these likely being ISO-8859-1 aka 'Latin-1'. As with UTF-8, 7-bit-safe ASCII remains unaffected regardless of the encoding family used.
The drawback to this encoding scheme is its inability to accommodate languages comprised of more than 128 symbols, or to safely display more than one family of symbols at one time. As well, ISO-8859 encodings have fallen out of favor with the rise of UTF. The ISO "Working Group" in charge of it having disbanded in 2004, leaving maintenance up to its parent subcommittee.
ISO-8859-1 is a legacy standards from back in 1980s (which has been abandoned in 2004). It can only represent 256 characters so only suitable for some languages in western world. Even for many supported languages, some characters are missing. If you create a text file in this encoding and try copy/paste some Chinese characters, you will see weird results. So in other words, don't use it. Unicode has taken over the world and UTF-8 is pretty much the standards these days unless you have some legacy reasons (like HTTP headers which needs to compatible with everything).
If you want to KEEP using ISO-8859-1 you can, but you will lose out on any benefits of a current, developing, character set.