Primers for PCR are usually designed to anneal perfectly to the target DNA and are small (~20nt) and not chemically modified. Primers for sequencing library construction and amplification are usually quite a bit larger because they are used to add on the 5' and 3' adapters (which bind to the sequencing flow cell and contain the recognition sites for the sequencing primers which are used during the actual sequencing-by-synthesis reaction.) The specific details depend on which library prep method or kit you're using - most kits will have a table in the user manual giving the full sequences of the primers/oligos used in the kit, so you can often figure out how they work by looking at all the sequences together.
Primers are primers. Laura's comments are correct.
Sequencing primers are usually additionally tagged with a fluorescent dye if not classical radioactive Sanger-sequencing is used (ask grandma how to). Because PCR should be highly selective the annealing temperature should be as high as possible to avoid unspecific amplification. For sequencing primers a lower annealing temperature can be used, especially if degenerated primers are used. We expect the motif of the sequencing primer to occure only once in the template.
However, for direct sequencing it is better to use sequencing primers that are a bit distant from the "dangling" ends of the PCR product.
You need two primers for PCR, folks often just pick one for the sequencing. Or set up two sequencing reactions, one from each direction, with one primer in each.
If you've cloned your PCR products into a common vector, you can use primers that anneal to the plasmid and face inwards towards your PCR insert. That's why a lot of sequencing facilities give you a choice of common primers (e.g. M13).
Scientists tend to go with "let's do what worked last time and not worry about it" when it comes to getting sequence data.