The difference is that there is no money motive at all in volunteer work. There are no paid ministers or missionaries so nobody is being potentially deprived of any income.
In many cases these persons wanting to server missions are in the USA illegally because their parents snuck them in years ago when they were kids. I fail to see how anyone is harmed by this.
No money motive?
The church gets free labor (which it apparently can't get without employing illegals), the illegals get further entrenched in this country, and you say there's no money motive?
First of all, can you guarantee that the illegals that the LDS Church is using won't get any welfare benefits, public schooling, or other social money? How are these illegals supporting themselves?
Second, even if that's true, what right does the Church have to flout immigration laws? What puts the Church above our laws?
What part of 'There are no paid ministers or missionaries' didn't you understand? It doesn't make one bit of difference where you are or where you are from, you don't get paid. Also, missionary work is something you volunteer for. You have to ask to be sent and anyone who asks and meets the requirements gets to serve.
Missionaries do not get welfare or other government money. They are expected to pay their own way, or get help from family or other members to cover living costs.
Lastly, the Church IS respecting the law. Asking for a change in the law is not flouting it.