A TV show is something that just airs on TV. A series is " a set of television or radio broadcasts on the same subject or using the same characters but in different situations " Definition from Cambridge dictionary, see sample sentences here https://dictionary.cambridge.org/dictionary/english/series

Hi there. Yes there is a difference. For example- "Friends" is a TV show with 10 series. So a series is just the collection of episodes, usually about 24 for a US drama or sitcom, but closer to 6 or 8 here in the UK for a drama. But the TV show is the actual thing itself- Broen has 3 series, The Big Bang Theory has 11 series, etc. As for synonyms...there aren't many, if any at all.