I'm going to have to disagree with the other responders. There are a variety of differences that could theoretically matter. Note that none of these are guaranteed, just that they COULD.
- Your adversary is probably also able to use the cloud and scale up the attack in a much more cost-effective manner.
- Some advanced adversaries could attempt something along the lines of a side channel attack. For example, if you are known to operate in a specific region or datacenter, they could attempt to choke out your available resources.
- As a DDOS victim in the cloud, you could end up experiencing the pain of the attack in two different ways: one, unavailability, but the second, increased cost of resources to handle the attack.
- As a DDOS victim in the cloud, you may be able to better cope with an attempted DDOS, because you may have easier access to resources that can scale with the attack. For example, you could more easily scale up your infrastructure, you are likely to have greater bandwidth, and you can more easily add new security "devices" into your traffic flows.
On the other hand, detection varies little. There's no silver bullet, no automatic answer. It's an active area for academic research. However, because you don't typically don't operate the full stack of your network infrastructure in the cloud, the cloud service provider has an interest in helping you deal with these threats. They may provide tools, documentation, or support to a bit to help with detection. Consult your provider, but as one example, AWS has some suggestions on how to use its tools for DDOS mitigation here: https://aws.amazon.com/answers/networking/aws-ddos-attack-mitigation/