So I have caught a couple of Sylesta, and I just tried to catch another. My net was gone, both my party animals got no experience, yet I do not see the new pet in my stables. Did I not catch it?

When you use a trap, three things can happen:
1. You catch the pet (it'll be written in the battle log, and there will be a link to the pet)
[trap is used in the process]
2. The trap fails, but the pet is still there for you to try capturing it again.
[trap is used in the process]
3. The trap fails. Scared, the pet runs away. The battle ends with no new pet.
[trap is used in the process]

Most likely, what you just encountered was the 3rd scenario, where you used the trap on the pet, but it fails and the pet ran away ending the battle with no pet or experience on your side. =/

Aha. Thank you for explaining that to me. I did not read the log carefully, so missed that the pet likely ran off.