Assuming you go with a reputable hosting provider, then no, there's really not much of a difference.
Theoretically, the possibility exists that some unpatched vulnerability in the host OS or virtualization software would allow shared peers to gain unauthorized access but today's virtualization solutions use protections built into the CPU to guard against this.
There was a demonstration of an attack which bypassed those hardware protections made a few months ago (google "VM side channel attack") but I haven't heard of it actually being exploited and I'm assuming the major VM vendors have mitigated that possibility by now.
You should still use software-based network and application firewalls if your security stance calls for it but those are as robust of a solution as their hardware-based counterparts.
So like everything else in security, it's always a cat-and-mouse game where black and white hats are trying to find new ways to exploit existing systems but unless you have very high security requirements most people agree that the security risk of hosting on a virtual server isn't appreciably higher.
Some hosting companies like Amazon for example have ISO 27001 Certification which indicates that their security practices were thoroughly audited by a third party.