King Albert knew his country was going to be crushed, but his honor forced him to fight. His son Leoplod II, was more pragmatic and surrendered when he thought all hope was lost. If Albert's character is different and he sees fighting as just causing needless bloodshed he might accept the German ultimatum with just a protest.
The British may still declare war though since they don't want Germany occupying the Belgian cost no matter what. Belgian wishes won't matter in London any more than they did in Berlin.
The important thing is this will save the Germans about two days, and maybe two corps of soldiers that would otherwise need to mask the Belgian Army in Antwerp and guard their line of communications through Belgium.
If the Germans have those two extra days they hit the French Fifth army on the extreme left of the French line and possibly envelop it while the BEF is still marching towards them. If the Fifth Army gets cut off and annihilated there is nothing Joffre or the Allied High Command can do. The French are going to get rolled up just as Schlieffen planned.
If the French avoid losing the Fifth Army the battle of the Marne may still play out differently or may never be fought at all. The BEF will not be able to link up in time and those two extra days and two extra corps may make the difference.