There is a number of benefits to pursuing both, the arbitration of your union grievance and a wrongful termination lawsuit (if there are grounds to bring such a claim). One of the main such benefits is the fact that losing one does not really affect the other, so in a way you get to have two shots of handling the same problem. This is because the two proceedings are fundamentally different.
First, a union arbitration takes place before an arbitrator and outside of court, while a wrongful termination lawsuit is pursued in court (State or Federal). More importantly, the standard of proof in both proceedings is very different. Generally, at a union grievance arbitration, the issue will be whether there was just cause for a discipline imposed against the aggrieved employee, and whether the discipline imposed was proportional to the alleged violation. In court, however, whether there was just cause for employee’s termination has no legal relevance, because a wrongful termination plaintiff needs to present evidence of discrimination or retaliation – to prove that he/she was either (1) treated differented due to being a member of protected class (i.e. age, religion, disability, familial status, gender, sexual orientation) or (2) suffered an adverse employment action due to engaging in a protected activity (i.e. complaining about unlawful discrimination or harassment, complaining about safety violations, engaging in political activities, reporting embezzlement of public funds, etc…)
Thus, if an employee loses at arbitration and fails to prove that there wasn’t just cause of discipline imposed, he can still be able to prove in court that the true reason for the discipline was discrimination or retaliation. Or, if an employee loses his case in court and fails to prove discrimination or retaliation, he still has an opportunity to prove that there was no just cause for discipline or tetermination.