On a 4-4 vote by the Supreme Court, the decision of the Appellate Court ruling would be allow to stand, but the Supreme Court would not be setting a precedent that other courts must follow. So it will depend on the case and what the Appellate Court which heard the case ruled.
Sometimes, like the climate change case, the progressive side would prevail. However, in others, like in the case of Texas' very tough anti-abortion law which was upheld by the lower courts, conservative opinions would be allow to stand with a 4-4 Supreme Court vote.
However, since a 4-4 Supreme Court vote on the Texas case wouldn't set a precedent, it would only apply for the case heard by the Appellate Court in the 5th Circuit which includes Texas. An identical anti-abortion law in say Arkansas might be struck down totally or in part by the Appellate Court in the 8th Circuit. So at least theoretically Appellate Courts in each of the 11 Circuits could issue completely different rulings in very similar cases and all of those cases would stand until the Supreme Court issues a definitive ruling on the matter.
When there are dissimilar rulings on the same issue of law in the lower courts, the Supreme Court almost always chooses to hear one of the case in order to settle the matter. but a 4 to 4 decision in the Supreme Court wouldn't do that - it would only settle that particular case. So in such situations, the highest court would most likely "kick the can down the road", waiting until a 9th Justice takes his/her place on the Court before taking up the issue.
There is usually consensus among the majority of Supreme Court Justices on most cases they hear so even with eight Justices, those cases will be decided with finality. It the highly ideological cases which have the most chance of being stalemated, but there are at least five major cases of that variety already on the docket for the upcoming session.
Until that ninth Supreme Court Justice takes his/her place on the highest bench and the Supreme Court is no longer powerless in deciding the most important case in the country, it is going to be a very interesting year.