Geologically, a glacier is permanent, wide, and thick, and it flows. The flowing is what causes crevasses to form. Topo maps will not necessarily distinguish between glaciers and permanent snow fields. Often if you look at the key, it will show the symbol and say something like "glacier or permanent snow field."
However, some old snow packs lying on sides of mountains could be tens of meters deep, and falling through a crack in one of these could mean very serious trouble as well, so why should it be any more safe to walk on snow fields than glaciers?
Not all glaciers have deep crevasses. However, the formation of crevasses is something that is specifically related to the fact that the glacier flows.
at what point do snow fields stop being safe to traverse alone with simply hiking poles and spikes?
Ask yourself how steep it is, how icy the conditions are, and how exposed you are. If you slip and fall in the conditions that exist, will you naturally slide to a stop in deep powder, or will you accelerate down a steep, icy 50 meter slope, hit a tree, and get killed?
If "spikes" means microspikes, then those are great for mixed conditions and for low angles and relatively soft snow, but on steeper, icier terrain you will want real crampons. Sometimes you don't know when you leave home whether the conditions will be icy enough to require crampons.
Trekking poles are useless for self-arrest. If you're going to be spending a lot of time in the mountains in snow, get an ice ax and get someone to teach you self-arrest.
Roping up is not just for crevasses, but that's a much broader topic that probably can't be covered in this format. For example, you'd use a rope if there is ice climbing or rock climbing involved in your climb.
Many glaciers do not have deep crevasses, and it's not necessary to travel in a rope team or be prepared with pickets and crevasse self-rescue skills. You can find out this kind of information from guidebooks, locals, or people who have climbed in the area.