Do spools of webbing still have splices?
Yes. As Paul's answerPaul's answer pointed out, there was a well-known accident caused by a climber purchasing webbing with a tape splice in the middle.
Furthermore, multiple retailers have various warnings that spools may contain up to three sections (ie, two splices) per 300ft spool of 1in webbing.
http://www.cmcrescue.com/equipment/tubular-webbing/
Note: Military standards allow for up to three sections per spool.
http://www.strapworks.com/Flat_Nylon_Webbing_p/fnw1.htm
For most webbing the maximum continuous length is 150ft, after which a new roll is started.
How weak is the splice?
It's just two pieces of webbing taped together. The tape's stickiness is unknown, so it should be considered to have an effective breaking strength of 0lbs (0kN).
However, in actual tests (slide 13) by Rocky Mountain Rescue Group, tape splices break around 25lbs (0.11kN):
Result 1 : Adhesive failure, tape intact, webbing pulled out, 20-25lbs (0.08-0.11kN).
Result 2 : Tape tore (see photograph), adhesive held, 24-28lbs (0.10-0.12kN).
How big is the section of weakness?
Presumably only the tape-covered portion of the splice is effected.
What do splices look like?
Like this: