What's the difference between "bush" and "shrub"? Are these absolute synonyms?
From a horticultural standpoint, 'bush' and 'shrub' are not synonymous. 'Bush' is a term used by the uninitiated to describe any bushlike plant, but in horticulture, bush usually refers more to the shape something makes, so you might see a plant description which says 'forms a bush' (as opposed to being tree like or with growth which doesn't bush out but goes straight up, for instance, Berberis 'Red Pillar'). 'Shrub', on the other hand, has a very distinct meaning - it is a plant which retains structure above ground year round, evergreen or otherwise, which cannot be split or divided because the growth is coming from one set of roots. Some shrubs can be considered small trees, but will still be defined as shrubs.
There are no exact synonyms. If there are two differently pronounced words that seem to be interchangeable in most instances without changing the meaning, there will still be contexts where one sounds more right than the other, one is more common than the other, one is not said in certain contexts, one is used differently in other contexts giving it different associations. Identical dictionary definitions do not make two words identical; the dictionary is not saying enough.
But, yes, despite the technically used differences by botanists and lawn care specialists, shrub and bush are very very similar in informal speech, so I'll try to show how they are different.
To me, a bush is more about the specific plant and shrub is more about the generic concept of those kinds of plants.
A shrub is "a woody plant smaller than a tree, usually having multiple permanent stems branching from or near the ground"
A bush is "a shrub or clump of shrubs with stems of moderate length." so they definitely seem to be exact synonyms.
I think of shrubs as decorative plants usually around or nearby a house. They are usually made of bushes (In fact I can't think of anything else the shrubs would be made of).
I suppose you could have bamboo as shrubs, and bamboo are definitely not bushes, though they can certainly be bushy (short and full as opposed to tall and thin).
You're more likely to point at a single plant and say 'that bush needs trimming', but to a whole bunch of them and say 'The shrubs need trimming'.
'in the bush' sounds a bit Australian or African, like you're out in the countryside with no trees. But that might also be called 'shrubland' or 'scrubland'. That's interesting question: "what is the difference between 'shrub' and 'scrub'?" (interesting because they are cognate like 'shirt' and 'skirt', or 'ship' and 'skiff')