Parker makes some good points. Just considering timing and productivity alone, these are my two cents…

For me, the everbearing types produce very little during the summer, they don't really get going until September. So, if you want to be harvesting a lot of berries in July, you should probably consider the summer bearing varieties. Also, everbearing varieties spread out their crops, so you rarely get very much at one time (unless you have rows of bushes). This means you might have to pick regularly and store them up over a few days or weeks to make that raspberry pie! Summer bearing varieties will produce a bumper crop all at once, which is good if you want to make a bunch of preserves or something like that.

Heruga:

For years I preached the benefits of fall bearing raspberries for my area (S.E. WI). No picnic beetles at that time, nice crops, easier to grow since mow the canes in early spring for a great fall crop. No worries about mice/vole/rabbit damage on overwintering canes.

No more. I now would not grow fall bearing in my area. The reason is spotted wing drosophila. Their population rises exponentially so highest in fall when the fall bearing raspberries are ripe. I now only grow summer bearing to hopefully beat the drosophila population. I suppose I could spray my fall crop but by fall I am tired of spraying since I also have 22 assorted fruit trees that I spray in spring/summer for good apple crops.

Summer bearing raspberries are now better for me than fall bearing.

If you want a big crop for canning, the summer berries are nice. I like both.

Many years ago I planted a patch of Mammoth raspberries -- thornless and supposed to be June-bearing. Every year I got a very light crop in summer, and then it loaded up for a huge crop in September, with the biggest berries. I attributed the oddity to my short growing season.

Spartan makes a very good point. The SWD female lays ~350 eggs. If the offspring all lived and reproduced a single pair could theoretically end up with 1.8 billion flies at summers end (Rutgers Cooperative Extension figures). Some raspberry growers just don't want that kind of protein in their berries.

I prefer a combination of summer (Floricane) and fall bearers (Primocane). The various varieties have unique flavor qualities that some people prefer over others. Although primocane varieties can be managed for both summer and fall crops, I am content to just mow those to the ground and only harvest a fall crop.