To begin with, like AIC and SIC, it is possible for trace tests and max eigenvalue tests to produce different outputs.
Unfortunately, which to use ("believe") is still under debate. Therefore, if you are writing academic papers, you need to report the results for both cases. Or if results are not that different, you may report the one case in the main body of the text and put the other in the Appendix.
The results can be different. Just check whether the result reveals a long term relationship exists or not according to both Trace and Max Eigen value statistics. The number of co-integrating rank might be different.