Cloud vs HCI: Pros and Cons

I think that the key points to consider are: security, performance, and CAPEX vs OPEX
Security: Having HCI on-premise allows you to keep your current security policies. For some customers having sensitive data on the cloud is not even an option due to their policies. If you go to the cloud you must remember that you are responsible for the security of your data, not the cloud service provider and new policy schemes may be needed.
Performance: You have to evaluate if the cloud provide the bandwidth, throughput, and availability that your operation requires vs. on-premise.
CAPEX vs OPEX: Even though there are some schemes that allow you to have HCI on-premise as EaaS (Everything as a Service like HPE GreenLake) most of the customers own their HCI infrastructure so depending on your expenditure convenience you will favor one or the other.

Cloud v HCI
In essence, Cloud 1st Strategy or whether to use Hyperconvergence Solutions does depend on the Customer Business Drivers moving forwards to the future, and then more importantly, analysis of the Applications, Data, and the Dependencies, so we can truly analyse :-

1. What Apps and Data can move to the Cloud – Including a true Analysis of Security and Data Sovereignty
2. What percentage of the Infrastructure needs to be On-Premise, across Multiple Sites, incorporating Disaster Recovery and Business Continuance
* There will be Data that is NOT suitable for Cloud Transition

So, for Apps and Data that conform to a NON-Cloud Strategy, I;e, they run On-Premise/Multi-Site/COLO, platforms such as HCI really make their mark around :-

1. Consolidation of footprint – Power/Cooling/Networking and Compute/Networking/Storage into a Node format which forms part of the HCI Cluster. Nodes can be added and scaled to grow the Cluster to keep pace with the Customer Business Needs. This is the case of SimpliVity, where INTEL XEON or AMD CPU Options/Nodes are deployed, a Cluster is formed to provide the HCI element, and the Cluster is built from the SimpliVity Nodes, and scaled to add on-demand growth
2. dHCI Technology takes this further by allowing Industry Leading commodity Servers to be used for the Compute Element, Networking products from the HPE Aruba/Mellanox/Cisco vendors to form the iSCSI element of the build, and finally high performance Storage, such as the HPE Nimble AF/HF Platforms to offer the Storage Tier.
* This type of HCI, allows the Customer to independently scale Compute to meet business needs, or even, scale the Storage without having to increase or change Compute. dHCI offers true Flexibility/Agility/Scalability

Finally, HCI Platforms such as the dHCI Technologies, have integration into HPE Cloud Volumes/AWS/Azure, so we can truly leverage Apps and Data requirements across On-Premise and Cloud

Hope this all helps.

There are several benefits of both Cloud and HCI that can be leveraged to the advantage of the feature rich HCI stack user hybrid style. The first is that many applications have not been designed for the cloud and require an on premise stack that can save data in the cloud and offer the same simplicity as cloud operations. If you select an HCI vendor that supports all of the Hypervisors and all of the clouds you can make your applications leverage each technology to your best advantage and lower OPEX costs by up to 60% without rewriting your applications to be cloud friendly. You can also simplify the entire stack and enjoy 5 microsecond latency and not make storage API calls that leave the kernel and introduce even more latency while they access storage from SAN and NAS devices. You can also serve up applications using FRAME technology with this stack that allows you to deploy solutions for remote workers in minutes that are fully secure. AOS offers full encryption and FIPS level 140-2 and better security built into the HCI stack right out the box, no need to go bolting on complex Frankenstein solutions like NSX that require several residents with deep knowledge of 8 different VMware stacks to operate the whole enchilada which increases your OPEX costs dramatically. AOS based HCI eliminates separate SAN, NAS and Object store silo's, it also eliminates system security and server with virtualization silo's and condenses them into one stack, so simple 8 year old children can administer it in a few mouse clicks. Mature HCI based also offers BC/DR benefits that will allow you to use the cloud for what the cloud os good at, BC/DR. Mature HCI vendors also offer their entire HCI stack for AWS and Azure so that you can drag Virtual machines from on prem to the cloud seamlessly. The San Jose based HCI vendor that does this is 4 years ahead of its competition (Dell-EMC) who only work with one Hypervisor while they work with any Hypervisor and all the cloud stacks concurrently. Nutanix Acropolis Operating System is the wave of the future and it runs on any hardware on their HCL from any server vendor. The HCL list is long. It is also a cluster based architecture that can be expanded one node at a time and they have GPU nodes as well. Nutanix Software Defined Valhalla is here today, so advanced everyone will think you are with the gods!!

HCI is on-prem, so it's simpler and easier to manage and integrate with applications and your network. Something like a Nutanix can give you a lot of functionality of the cloud without having to deal with the massive headache that is designing your network and applications to be able to utilize the cloud effectively (for Infrastructure). SaaS is a fantastic use of the cloud, but infrastructure-as-a-service hasn't matured in process or manageability yet to justify. It will always cost more to be in the cloud, and it will always be more difficult to get to it.

The cloud is amazing if you use it right.

HCI is on prem and requires support to keep it running. There are times when there may be an power outage where it may be difficult to the get HCI up and running again after a crash. HCI is costly, some 3 node clusters nearing $100k, but HCI is also available below $50k. Cloud is pay as you go, and cloud can grow with the end users requirements. However, over time, the cloud (opex) may exceed the cost of the HCI (capex). The other issue is on-prem (HCI) should have better performance as it is in-house and the end user is in total control of the security,

The main benefit is about the elastic and fast deployment of the cloud vs the time with purchase and deliver of HW for HCI.

Why Cloud vs HCI?
The best option for Cloud is a hybrid Cloud and the best option (on premises or cloud) for performance, resilience and so on is HCI.
If the demand analysis points to Cloud: It should be addressed whit Cloud. Otherwise, if the demand analysis points to HCI: It should be addressed whit HCI.

I think there is no one answer for this question
You may go both HCI and cloud together because it depends on the services you want to utilise
Sometimes the cloud is more expensive than HCI like if you want to design a 10 server each one with minimum of 128 go memory and 12 or more core it will cost you a lot in the cloud As Capex or Opex
But for like high transaction web services it will cost you less specially the cost of internet and high availability you will consider for this web site