When it comes to fall-bearing varieties, ripening labels refer to the later primocane or "fall" crop rather than the "summer" floricane crop, and people further north should prioritize those that ripen early since the fall crop of the others might start too late to produce enough to be worthwhile.  You may want to prioritize early ripening summer-bearing varieties as well, but this trait is not just beneficial for those in the north.  Their crop tends to finish before significant numbers of the spotted winged drosophila emerge, which is an invasive fruit fly from Asia that can pierce the skin of slightly underripe fruit if it is thin enough (like raspberries).

Summer and Fall Bearing

Summer-bearing varieties will produce a large crop earlier in the season on second year growth called floricanes.  Fall-bearing varieties will produce a crop later in the season on first year growth called primocanes, and in some cases, this crop can be spread out over a period of months, if your grow season allows it.  A few summer-bearing varieties may produce a small crop on their primocanes, and that causes some places to refer to them as fall-bearing.

Climate

Raspberries are better suited for climates with fairly cool summers, but at least some varieties can handle more central parts of North America or the more northern part of the humid subtropical climate.  When testing their limits, the summer-bearing floricanes may struggle to produce in the heat while the fall-bearing primocanes perform just fine.  If this happens to be the case, you can help fall-bearing varieties save energy and shift production to their primocanes by cutting the plant down every year.  In harsher climates further south, you are better off growing disease resistant blackberries.

Pests and Disease

Raspberries can suffer from a number of afflictions, but most of them do not seem to be too common or significant for home-growers.  The spotted winged drosophila is the most complained about, but as I mentioned above, you can avoid problems with them by planting early ripening summer-bearing varieties.  Cane borers are also quite common, but they aren't too difficult to control since their presence can vary considerably by year or take time to build and, in at least two out of three cases, are easy to manually remove.

The Redneck Cane Borer will create a gall that is generally located on the lower half, and the entire cane should be removed once noticed.  Gall-like structures found at ground level may be caused by the Raspberry Crown Borer instead, but this species is more difficult to manage, since it feeds on the roots and crown as well.  However, limited trials indicate that the nematodes Steinernema feltiae and Steinernema carpocapsae can reduce their population by one to two-thirds, and mass trapping with a lure may be enough to keep them within manageable levels without additional effort.  The Raspberry Cane Borer will puncture two rings near the top of your primocanes, which then causes the area above it to wilt.  This occurs around early summer, and if you do not wait too long, cutting a few inches below that point should be enough to remove the larva.

Most varieties have some susceptibility to phytophthora root rot, a disease that can quickly kill the plant, but this is only a problem in wet climates or soils that do not drain well.  In such conditions, you can help prevent disease by planting your raspberries on a raised bed, but a more reliable option would be to prioritize those that are known to have some resistance.  Anthracnose of the cane can also kill black raspberries in wet climates, but red and yellow raspberries are more resistant to this disease.

The Raspberry Bushy Dwarf Virus (RBDV) can cause a number of symptoms such as crumbly fruit, but the presence and severity of these symptoms may depend on the co-infection of other viruses.  Additionally, the lack of complaints imply that they are more of a problem for commercial growers and those with native raspberries nearby, but I believe the most vulnerable are being phased out as well.  Red, yellow, and purple raspberries, as well as blackberries, may also harbor viruses that are dangerous to black raspberries while showing no symptoms themselves, so the best practice would be to put some distance between them if you have the space (with black raspberries being upwind).