When you are using Full Tunnel VPN (which is the default setting) you will have a couple of things that you need to configure on the ASA.
First, the ASA by default won't allow traffic to enter through an interface and then leave through that same interface. This is what essentially happens when the traffic from the VPN Client comes to the ASA and then heads out to the Internet. In your case the traffic comes through the "outside" and leaves through the "outside" interface.
You will need this command
same-security-traffic permit intra-interface
You can check if its enabled at the moment with the command
show run same-security-traffic
Second, the VPN users will need to have NAT configuration just like any LAN users behind the actual ASA. So you will essentially have to configure Dynamic PAT for traffic from "outside" to "outside"
You can accomplish that with the following configuration
object network VPN-PAT
subnet 10.4.1.0 255.255.255.240
nat (outside,outside) dynamic interface
I would imagine that this should do it for you to be able to connect to the Internet and to the LAN network when the VPN is active.
Hope this helps
Let me know how it goes.