Whilst the rulers Kaiser Wilhelm II and Tsar Nicholas II generally (although it should be said not always) shared a warm relationship with each other, that was only one small factor within a web of reasons that the relationships between these states ended up how they did leading up to World War 1. Whilst I'm by no means an expert in the matter, I have been reading up on this recently and can provide some answers and examples. Please note that this is not meant to be an exhaustive answer, which would be almost impossible to provide due to the notoriously difficult to decipher web of connections between the great powers during this period.

During the years prior to WW1 Russia went through troubled times, such as losing the Russo-Japanese war (along with a significant proportion of its fleet) in 1905, which was followed by a thwarted revolution that same year. This has a significant effect on its finances and it was France that stepped up to help its fortunes through a series of massive loans.

These loans were provided not out of good will, but through a desire get the Russians on side. This was seen clearly at the Algeciras Conference in 1906, where the Russian delegation supported the French over German arguments in return for the promise of further loans.

The French were well aware that their chief enemy in Europe was Germany, and were still sore from ceding Alsace and Lorraine to the German Empire following the Franco-Prussian war in 1870. Through keeping the Russians in a defensive alliance the French hoped to deter further German aggression, due to the difficulty that Germany would have fighting a war on two fronts. The French were all too keen to make this difficulty felt more keenly, funding through loans such infrastructure projects as modernising railway networks in the west of Russia, enabling the quicker mobilisation of troops from the Russian interior to the German border in the case of war.

French interests and finances weren't the only influencing factors. In fact, the Germans and Russians held a top secret treaty known as the Reinsurance Treaty up until 1890. This stated that in the event of either country becoming embroiled in war with a third great power, but that this would not apply if either Germany attacked France or Russia attacked Austria-Hungary. However, following Otto von Bismarck's dismissal in 1890 this treaty was not resigned (seemingly through a combination of the German's viewing it as being too strongly in favour of Russia and sheer incompetence of remembering to keep it active at the appropriate time). With the cancellation of this treaty leaving Russia without much-needed allies (and understandably quite annoyed with the Germans), they went on to form their alliance with France.

Another factor in the growing unfriendliness between the German and Russian empires was their relationship with Austria-Hungary. Historically and culturally connected to the German Empire, Austria-Hungary was importantly one of Germany's only friends on the continent. Germany did not want to be diplomatically and militarily isolated on the continent and so viewed their relationship with Austria-Hungary as important to maintain.

(It is probably a good place to note here that whilst Germany was also allied with Italy in the Triple Alliance during this time, Italian relations had been cooling with Germany and warming with France for a period before WW1. The Italians were not viewed as a major power at this point and so Germany did not think their alliance with them as of much importance as it viewed its alliance with Austria-Hungary).

This was at odds with the relationship of the Russian Empire with Austria-Hungary, who were in almost constant disagreement over issues in the Balkan states. Indeed, here would be where the final trigger for WW1 was pulled. These conflicting feelings towards their neighbour was another source of friction for the two empires.

I'm going to stop here as I feel that I've answered your question with enough examples of how German and Russian relations were so much more complex than merely the relationship between their respective heads of states. These are just a few examples that came to mind and I'm sure there are many others that are relevant to raise, perhaps by someone more knowledgeable than myself. I'd recommend reading the source I have provided for a fascinating look into the relationships between states prior to WW1.

Sources:

Margaret MacMillan - The War that Ended Peace: How Europe abandoned peace for the First World War (2013)

It's perhaps not as much of a surprise as it might seem, although it did provide the oddity of one of Europe's most autocratic states allying with its only big republic.

One important thing to note is that geopolitics had changed a lot since the 1850s. France had been its own Empire T that time, and one of the biggest continental powers, while Germany was a loose confederation of independent states, with Prussia and Austria being the biggest, and on competitive terms with each other. This changed massively after 1871, when Prussia unified all of non-Austrian Germany, inflicted a severe defeat on France, and began a process of intensive industrialization. It also put itself at the heart of a number of European alliances through Bismarck's work, including the Dual Alliance with Austria-Hungary, which was a mutual defense treaty. This treaty remained in force right up through World War I, and was the justification for Germany's entry into the war on Austria's side.

Bismarck also hoped to involve Russia in this alliance system (the main goal was to isolate France), and as such involved Russia in the Three Emperors League, set up in 1873 and running until 1887. This wasn't so much an alliance per se as an established forum for Germany, Austria and Russia to maintain their respective spheres of influence in southeastern Europe, and assure that any changes there were mutually agreed upon. Eventually Russian and Austrian competition with each other undermined this system, and Bismarck replaced it with the secret Reinsurance Treaty in 1887, where Germany and Russia guaranteed neutrality in each other's conflicts with other states, and Germany recognized a Russian sphere of influence in Bulgaria. With Bismarck's dismissal in 1890 this treaty was annulled by Germany - it wasn't considered a particularly good deal by its opponents.

Just as an aside: this should point us towards what Russia's interests were in this period. It wanted as much influence in the Ottoman Empire as it could get - either indirectly or through direct annexation. The dream of course was to control "Tsargrad", aka Constantinople, as it controlled the Turkish Straits and Russian access to the Mediterranean (and warm water all-year maritime trade routes). Connected with this was an interest in protecting (or controlling) minority groups in the Ottoman Empire, justified either through Russian Orthodoxy (the Crimean War was actually precipitated by a dispute between France and Russia over who would have protective rights over Christians in the Ottoman Empire), or by Pan-Slavism, the idea that the Russian Empire should seek to unite Slavic peoples under the star's rule. Bulgaria in particular fit nicely into this perceived sphere of influence, as it was fighting for independence from Ottoman rule, was Orthodox, and was Slavic. Serbia was the other big contender. Of course, since Serbia and Bulgaria ended up in conflict quite a bit with each other, it meant that one tended to seek support from Russia and the other would then seek support from Russia's regional competitor, Austria. This local alliance system flipped between 1877-78 (when Russia fought Turkey on Bulgaria's behalf and was opposed by Austria), and World War I.

Anyway, to bring France into the equation. When Russia's alliance system with Germany dissolved, France was able to step in - it was less that Russia was looking to fight with Germany, as much as it was seeing potential conflict with Germany's Austrian ally. France, however, was planning for a potential fight with Germany, and saw Russia as a massive counterweight. France reached out in 1891 with a naval visit to St. Petersburg and managed to negotiate a "consultative pact" that ultimately led to a formal alliance in 1894, which would last until 1917. So in very specific terms, Russia had a military alliance with France in 1914 where it did not with Germany, while Germany did have an alliance with Austria, who was an increasingly fierce competitor with Russia for influence in southeastern Europe.

On top of this, the 1890s saw massive investment by France in Russia, which spurred that latter country's own industrialization from that time (especially in railroad construction and armaments). Out of some 63 billion rubles' worth of public debt held by the Russian government in January 1918, 44 billion was domestic and 19 billion was foreign, and France was the biggest source of that foreign debt. This was to the point that Bolshevik repudiation of tsarist debt was the major stumbling block with re-establishing relations with debtor nations, with France being the longest holdout (and there are still French inheritors of tsarist-era bonds seeking legal restitution). ETA of particular interest would be the Anglo-French loan of 1906, which totaled some 900 million rubles, and was a joint loan by French and British banks. It was negotiated by Russian Minister Sergei Witte, and helped Russia reorganize and rearm after the defeat in the Russo-Japanese War and the 1905 Revolution, and was a diplomatic coup in that British banks participated in the financing (the British government saw this as a means of keeping German lenders out of Russia).

So: France and Russia saw each other after 1891 as natural counterweights to their respective competitors, Germany and Austria, who happened to be in a strong military alliance. Russia in particular saw Austria as a threat to its interests in southeastern Europe. France was also playing a crucial financial and economic role in Russian industrial modernization. These were all reasons that drew France and Russia together into a formal alliance that was activated by the July Crisis of 1914.

While more can be said, I gave a synopsis of every nation's reasonings for joining in their respective alliance in this answer over here, including France and Russia.