Bamboo is a grass. Cane Palms (probably Arecas) are, well, palm trees. Neither are invasive, however, the running bamboo will spread faster and further than an Areca and are messier. However, Arecas require considerable maintenance - picking up the dead fronds, cutting down the seeds before they fall and sprout new ones.

If it's an Areca palm they can get fairly wide and messy unless you pick up the dead fronds and fruit. Depending on the location they can get tall but (in my humble experience) I've seen them mostly get wider. I'm in Central FL though.

As far as invasive, running bamboo can get out of hand, much more so than clumping, but neither is really invasive. You just have to keep up on them :)

I planted some Arecas on the south side of my house for privacy 3 1/2 years ago, I wish i had planted them on the west side...so if I get clumping bamboo I can control how wide they get. I only have about 4 feet from my fence that they can grow or my hubby will have a fit if he can't get the mower between the bamboo and the shed to get to the back gate. Wow, that was a mouth full, sorry.