Empathize phase of design thinking usually needs a user research. But are there any particular differences in terms of analysis tools & techniques, research methods etc for the user research done during design thinking when compared to that done during ux design?
I would say that the type of methodology, tools & techniques used in research (regardless whether you call it design thinking or ux design) should be determined by the type of information you want to collect and not necessarily the approach that you take.
So, if you want to obtain information about what a person thinks you can pose questions to them in person (this could be an interview or as part of doing contextual enquiry), or you can send them a survey to fill out and many other methods. Of course, each method has different limitations and benefits, so you have to weigh up what the trade-offs are given your resources such as the ability to actually travel to their location and do the interview in face-to-face.
Conversely if you want to obtain information about what a person does, you can also ask them what they would do, or observe them performing the activity (this could be user testing or as part of doing contextual enquiry), or you can send them a survey to fill out and many other methods. Since there is usually some difference between what a person says and what does, it is advisable to validate what someone says by observing what they do, and also to understand the context of what they are doing by clarifying their intent in performing an action.
Hopefully you can see that it is possible to capture the same type of information but the degree of confidence that you assign to the results may vary due to the method used. For instance, observing someone performing a task may be more reliable compared to a survey response, but having someone watch you perform a task might also introduce some changes to how someone normally does something. Then there are some ethical concerns around recording or capturing user data without them knowing that it is taking place.
This is the difficult of user research - ensuring that the information you have collected is not subject to your own or the user's bias when providing the information. Therefore the analysis and synthesis of results from user research is both a science and art.
I agree with what Michael had written previously. I would make an additional point: User Experience is a field; Design Thinking is a methodology, an approach. In that sense, you cannot really state your question as UX may include DT in a specific application. DT is often used to quickly iterate the design and decisional process and to enable the collaboration among different stakeholders. In this context, the user research might be done in a 'quick and dirty' fashion. A UX specialist not adopting DT will consider the best research method and the appropriate timings that suit a specific project/need. Both UX and DT grab their methods from social sciences. Their application is affected by the type of project.
Tbh, I see DT as another name to refresh the old 'iterative design' or the new 'circular design'. It is just a new way to use the same tools to better approach specific business cases.