Are the Finesse APIs (REST & JavaScript) different for Finesse with UCCE vs. UCCX?

The Finesse APIs, both REST & JavaScript are the same between UCCE and UCCX, but there might be some slight differences in the response/output. These differences are documented in the Finesse Developer Guide. 

The differences between the two platforms fall into two categories: 

1. There are a few things that work on one platform but not the other. For instance, currently, QueueStatistics is supported in UCCE, but not UCCX. The API request will still work for UCCX, but the values of the response and notifications will be -1.
2. There are some subtle differences in state models between the two platforms. For example, in UCCX while on a direct dial call, the Agent's state is "NOT_READY", but in UCCE, the Agent's state is "TALKING". There are not many of these, but if you are familiar with the two platforms, then you probably understand these already.

If needed, you can use the deploymentType of the SystemInfo to determine which platform the gadget/application is running on and do the necessary changes accordingly.

Can gadgets developed for Finesse for UCCE be used on Finesse for UCCE without any change or major change?

Yes, as long as you understand the differences. If you code your gadgets in a very clean way, many of these differences should not really affect you. Of course if you try to utilize an API that isn't present in both platforms, you might get values of -1 or something not meaningful. When it comes to the state model differences, just handle the states as they come. Gadgets are loaded in the Gadget Layout in the same way across the two platforms.