I'm not a fan of stone/gravel mulch. I use it in one spot for a specific reason but it's hard to keep it looking clean. Weeds will still grow in it or past it. Either because the landscape fabric underneath deteriorates or because organic material falls onto the gravel and collects in the spaces between the gravel and allows seeds to germinate.
If you want to clean it well you basically have to set up a sifting operation like an 1850's prospector. Since it doesn't decompose topping it off with fresh gravel just creates a deeper mulch bed. As the landscape fabric wears out the gravel can sink into the soil so when that happens I guess you can top off then.
It doesn't really suppress weeds as well as organic mulches. It's mostly the landscape fabric that does that. If you ever want to change it it's a giant PITA.
The microclimate around the gravel bed will be warmer and retain heat longer which may cause issues with some plants or require more frequent watering.
I'm not a big fan of generic wood mulch either. For wood mulch I prefer cedar mulch because that's one of the better choices available around me. The oils in it can repel insects such as termites, cedar mulch is less likely to harbor artillery fungus (as does cypress and pine park mulch), it lasts a long time, I don't have to worry as much about scrap pressure treated wood or toxic paint winding up in the mulch and I like the smell when I put it down.
A 1/2 cubic foot bag of gravel costs about $4. For that same $4 I can get a high quality 3 cubic foot bag of cedar mulch. It's lighter, easier to work with, suppresses weeds when put on thick enough, breaks down to add nutrients to the soil. Insulates the soil and retains moisture and I like the way it looks.
You put on about 3" the first year, then add about 1" every year, or even every other year depending on your climate. After a month or so the color fades to grey. You can get died mulches but the color only lasts about a season and will need to be topped off to look new or you can get some mulch dyes to refresh the color.
Haven't had an issue with either washing away but the cedar mulch seems to hold better on slopes.
Aesthetics... when done properly stone mulch can have a very dramatic and high end look. When it starts to look like crap then it looks like crap. Organic mulches are easy to refresh and have looking like new by only topping off a bit every year.
Costs for a 100 square foot bed with 3" of mulch:
Stone: Initial cost 50 bags * $4 = $200 though 50 bags is a lot to transport bulk pricing would be cheaper.
Cedar: Initial cost 9 bags * $4 = $36
Top off with 3 bags every year = $12 a year.
It would take over 12 years for the cedar mulch to cost as much as the stone. Are you sure you're not going to want to rearrange your landscape, want to change color scheme in mulch or will still like the same look for such a long period of time?
Spreading around some extra mulch on top of beds to me feels a lot easier than trying to clean gravel beds.