Hi WesternAmerican,
Yes, I would call them synonyms as they express the same broad concept (of something getting wrapped up or made to be more difficult than before). But the usage is different (collocation) depending on the situation and other words or ideas you want to use them with.
When I read your "to entangle things" I get a sense of things getting tied together physically or in a messy way.
When I read "complicate things" I get a sense of an intangible (something you can't touch) idea that has become more difficult and or confusing than before.
So I think you could get your dirty clothes all entangled in the laundry while you are trying to load them into the washer, but you could complicate things for yourself if you then decided to start sorting the clothes into whites and darks while they are all entangled there in the washer.
One (entangle) is more common for physical objects, while the other (complicate) refers to problems or situations (things we usually can't touch).
Does that make sense? I hope so!
Best,
Jason