None. In fact, Rust software will typically include larger and more frequent downloads for bug fixes, upgrades, and security fixes. The reason for that is that C programs, like vim, can fix bugs and security issues by updating individual dynamic libraries, which are likely to be part of the system. Rust programs can't, because they are statically linked, and this means the whole program needs to be downloaded and installed again. Also, Rust makes it much easier to manage and include library dependencies, which has the consequence that Rust programs in the long run will have an exponentially larger and deeper tree of dependencies, which means that fixes which are needed to fix some bug in a library will be more frequent. This is an effect which can be observed well for other languages like Python or Javascript with npm libraries.
That is probably fine for a web browser (which you basically need to re-install every other week anyways, and which uses auto-update for that), and might not have too much impact for a small tool, but once a program accumulates a large set of functionality and complexity (and therefore has more dependencies) let's say like Gimp, OpenOffice, VLC, or FileZilla, this is likely to become noticeable.
Another thing which might or might not be an issue is that software which is very fast-moving, and in rapid development, and adopts to breaking changes in its dependencies (as the Rust development model supports) is also a bit more likely to breaking changes in its own user interface, APIs and features. The reason for this is that with the included bundling of pinned library versions into the Rust executables, there is much less pressure to keep a stable API, as when software is composed of individual components which are used in many places with a strictly-defined API. This is actually not necessarily the case, it depends on what the software's authors promise (and keep) to its users. And breaking changes are very often an inconvenience for users; users and developers of software tend to have somewhat differnet opinions on how much friction and breakage is OK in order to get upgraded software.