Almost interchangeable terms really. Herbaceous perennials are ones which die back to a resting bud or series of buds over winter.
Perennial just means living for more than one or two years.
Plants which grow and flower and die in the same year are annuals. Plants which grow one year and flower the next and then die are biennials.

Perennial means it doesn't die over winter, but might mean it remains evergreen, or visible above ground in winter; adding 'herbaceous' in front usually means it disappears more or less below ground in winter, and regrows the following year.

Technically, "perennial" means any plant that lives three years or more, which would also include shrubs and trees. When a gardener refers to a perennial, they mean an herbaceous perennial, which doesn't have any permanent woody structure above ground. In warmer climates, such as where I live, some herbaceous perennials are evergreen, such as Liriope, Pampas Grass, or evergreen Daylilies, and some grow in the winter, and go dormant in the summer, such as many Penstemons. In colder climates, such as the UK, herbaceous perennials generally grow and bloom in the summer, and die down to ground level in winter.

Erm, its not strictly true to say gardeners mean herbaceous perennials when they use the term perennials, Tug, the phrase 'herbaceous perennial' in the UK specifically refers to plants which disappear in winter. I tend to use the terms evergreen perennial or herbaceous perennial to differentiate - there's a good explanation here

https://www.rhs.org.uk/advice/profile?PID=868

I was speaking from the botanical and ecological point of view when I spoke of perennials including trees and shrubs--also bulbs, though they are actually a special case of herbaceous perennial--while the RHS is speaking from the UK gardeners' point of view. Among U.S. gardeners, evergreen perennials, such as Bergenia, are included among herbaceous perennials, since they have no permanent woody structure above ground. To save verbiage, we gardeners just call them all perennials. More of a language difference, than anything else.

For sure, I realised that in the first place, Tug, its another of those differences in terminology between USA and UK/Europe

Hi, it's as Winston Churchill once said, the british and the Americans are 2 countries seperated by a common language, Derek.

botanically speaking herbaceous means lacking in woody structures, ie the stems do not undergo secondary thickening [becoming woody] they dont always die back fully as bergenia is a good example.

I tend to go from a botanical view and hadn't realised there was a 'gardening term'.

lots of confusion with evergreen too as we know due to all the questions we get on here :o))

It differentiates between plants like Campanula muralis, Ajuga reptans and others which remain above ground and green all winter, yet are not shrubs and have no woody parts, and other plants like Rudbeckia and Dicentra, which completely vanish.

Generally, botanical terms are carefully defined, and published in peer reviewed literature. Gardening terms are defined mostly by common usage--which is perfectly valid, but allows for differences between the U.S., and the UK.