Statement of Interest for Ph.D. admissions
The primary difference between a statement of interest (or purpose) and a thesis proposal is the specificity. A thesis proposal outlines a specific project to be pursued during the thesis research, and will usually be written at some point during the process. However, in many cases, this is done after the graduate student has been accepted and joined a particular research group, rather than before admission has been granted. When asked for at the admission stage, it's usually in cases where the assignment to a particular research advisor is done at the outset of the doctoral program.

A statement of interest, on the other hand, simply details the kinds of problems or research areas a given student would like to pursue as a graduate student. This is usually asked for in admission cases where the decision is being made at the departmental level, rather than the individual faculty member level. A student is not specifically committing to a particular project at this stage—nor does the department necessarily want such a commitment.