Solving Budgeted Maximum Coverage Problem using Greedy and Genetic Algorithm
Absolutely they can. The approximation algorithms give a formal guarantee that the solution won't be too bad and they quantify what this means whereas with a metaheuristic such as GA all bets are off. Really, the only guarantee you have with GA is that the solution is valid, but that's it.
Your best bet is to just implement the GA and give it a try. Much will depend on the details and choices you make inside the method as well as on your instances and their structure.