How important were crowns as a symbol of power and were they passed down through a dynasty? I’ve always thought that crowns would hold great significance to both the small folk and nobility of a monarchy, and would be held up as evidence that the current ruler has a physical link to a line of previous kings (which the Iron Throne may do for Westeros). I’ve always found it strange that Renly so willingly wore a new crown blatantly featuring the Tyrell heraldy instead of wanting to claim Robert’s.

Crowns were very important as symbols, and they often were passed down, even across dynasties. 

However, it was very rare for monarchies to only have the one crown, because the oldest crowns were usually too valuable and heavy to be worn every day.  The result is that you get multiple crowns and thus multiple symbols, which GRRM’s notes on the Amok portraits of the Targaryens does a good job of laying out for the Targaryen dynasty.

I’m sure that Renly would have loved to have Robert’s crown to use, but that was never going to happen base on how Robert’s death went down. However, I don’t find it strange at all that Renly combined the Baratheon heraldry with the Tyrell heraldry:

The slender circlet around his brows seemed to suit him well. It was soft gold, a ring of roses exquisitely wrought; at the front lifted a stag’s head of dark green jade, adorned with golden eyes and golden antlers.
The crowned stag decorated the king’s green velvet tunic as well, worked in gold thread upon his chest; the Baratheon sigil in the colors of Highgarden.

For all his faults, Renly is someone who understands political symbolism, so this isn’t an accident. Rather, it’s a bit of quid-pro-quo to reassure the Tyrells that Renly is the path to a Tyrell on the Iron Throne, and to reassure the lords of the Reach that swearing allegiance to Renly is like swearing allegiance to the Tyrells, and you’ve already done that, so everything’s copacetic.