A 3 term or President "for life" tradition may mean that AJ never becomes President; the butterflies involved here may mean that none of the OTL men who have served as POTUS since the mid 19th Century would rise to the office.
Indeed, the ramifications of Washington serving Adams' term are far reaching and begin before concerns about the 2 term tradition. [Though, I'd add, that tradition is certainly important in the long run. The biggest difference IMO is whether Washington gets a chance to voluntarily give up power or whether he dies in office.]
Washington deciding to serve a third term probably needs an outside POD. Such a POD may involve the Jay Treaty, in which case there's a threat of war with Great Britain.
Washington deciding to run again (without considering the idea that Washington would probably not be inclined to do so unless the circumstances of 1796 are somewhat different) may confront Thomas Jefferson with a bigger choice about whether he will return to politics. If Washington runs again, John Adams might fail to get enough votes to be Washington's VP. However, TJ might also decide to pursue other means of confronting Hamilton and the Federalists, perhaps by running for the Senate or House. This is huge because changing the nature of the Federalist / Democratic-Republican face off will hugely impact how American politics works -- you could very easily end up with the Presidency begin gradually undermined in one fashion or another (remember, all of this predates Clay's committee system).
Furthermore, Washington might very well take different actions vis-a-vis the Quasi War (or be forced to be different events). The US relationship with GB may already be substantially altered. This will have big impacts because it may set the stage for very different conflicts later on.