You have 5 types of overlapping letters to that can be sent to a medical admission committee
Updates: new info you want to make the committee aware of. The information should be worthwhile and significant. Some schools encourage these buy many schools discourage or list policies not accepting updates.
Thank you: after interview
Letter of Interest: Typically, a post-interview but pre-decision letter that can reiterate and reinforce the school would be a good fit, usually with information you learned from your interview. This usually does not include a promise to attend if selected.
Letter of Intent: Typically written by applicants who have been accepted by one school but are on WL at another preferred school, outlining the reasons why you prefer it. There is little to scant evidence that these are particularly effective. It should be noted these are thought to have even less impact for those who are just on WL.
I will add one other type that is rarely used any more, the Reconsideration Letter: This is an applicant rejected pre-interview who has a large anomaly in their academic background that can be viewed through mitigating circumstances and asks to be reconsidered. With the non-screened/minimally pre-screened secondaries, there are plenty of opportunities for an applicant to express these circumstances.
To answer your specific question, unless, you have something substantial to say to the committee that was not originally mentioned in your application or a worthy update, an Letter of Interest, particularly before an interview is mostly useless. Indeed, many applicants write them sound more desperate than interested and that will work against you.