Intentional cross-border mobility for these services is different. The Court of Justice has promoted mobility of students with equality of nationals of the member states (judgment Gravier). They thus enjoy education at expenses of locals. Mobility of patients promoted with exportability of reimbursements has been based on free movement of services and goods (Kohll and subsequent).
The second approach complies with model of economic integration and with separation of welfare systems more than the first one. Widespread resort of patients would boost providers in host country and push their home states into improvements, while unlimited student immigration could exhaust host country and promotes free-riding.
Paradoxically, acceptation of the two approaches is opposite. Student immigration is perceived as enrichment. Reimbursement of treatment found abroad was opposed by the member states fearing destabilization of their healthcare. Despite the Directive on Patients’ Rights, poor implementation reveals continuous efforts to curtail its impact.
We can hypothesize that different approaches stem from different models of solidarity. Healthcare was always perceived as individual benefit. Education is perceived as public good.