Cisco VPN client is for connecting to an IPsec-based remote access VPN.
AnyConnect client (whether on mobile device or PC) is for connecting an SSL remote access VPN. Depending on your current licensing, you may also need additional licenses on the ASA (AnyConnect Essentials and AnyConnect for Mobile (old style) or AnyConnect Plus (new style)).
The SSL VPN type requires a certificate on the ASA or router headend.
You can use a self-signed certificate; but it needs to be created, bound to the outside interface and the group policy needs to have SSL enabled.
No certificate is required on the client - it only needs to trust the ASA's certificate. (That's why we recommend certificates issued from a public root CA.)