To my (skin-deep) understanding of the protests and revolutions of both years, there appear to be quite a number of similarities. Both were revolts by masses of people against the established order of the time. Both transcended borders, though the former was confined to Europe while the latter was global. Neither forced any sort of political change in the moment (as far as I am aware), though I have heard arguments that the impact of both reverberated for the next several decades. Is that picture correct? Regardless, can you draw connections between the two movements or are they wholly different?
I tend to think of 1848 as the birth year of many revolutionary trends, while 1968 was in fact the death year of those revolutionary trends. Both tumultuous, but very distinct.
Here are my two favorite books on each:
1848: Year of Revolution by Mike Rapport
1968: The Year that Rocked the World by Mark Kurlansky
First, let's look at the years before 1848 quickly. In 1815, Napoleon Bonaparte of France was defeated at Waterloo and the ancien régime was restored to Europe. The Bourbon dynasty was installed again in France, and the Congress of Vienna sought to restore a balance of power to continental Europe to prevent the kind of bloodbath that the Napoleonic Wars had unleashed.
The years between 1815 and 1848 saw the rise of several new political philosophies, if not ideologies. In France, you had continued republican agitation and radicalism, and nationalist ideas in France continued to be expressed in the context of republicanism. In the German and Italian states, both of which had some semblance of unity under Napoleon's empire, you had nationalism focused on uniting a country, whether under a monarchy or a republic - this was literal nationalism, in that there was a people speaking a similar language who were not citizens of a unified nation-state. Finally, the advance of the industrial revolution brought profound changes on the life of ordinary working people, and socialism began to emerge, although at this point, only loosely defined and representing anything from libertarian socialism to communism to syndicalism to anarchism. These movements were all largely prototypical of later movements. Often "classical" variants of liberalism, socialism, Marxism, anarchism and the like refer to later 19th-century ideological descendants, to the point where we might best describe the ideas and ideologies of 1848 as "pre-classical".
Events cascaded across Europe throughout the year, beginning with a Sicilian revolt to throw off Bourbon rule in January, followed by a February Revolution in France, a March Revolution in the German states, a revolution in Denmark that ended in a constitutional monarchy, national revolutions across the Habsburg Empire, including the largest of such in Hungary, followed again by revolts in Switzerland, Poland, Belgium and Ireland.
The results were mixed, or lead to outright defeat.
The July Monarchy was overthrown in France and the Second Republic was declared, but only to be overturned in four years by Napoleon Bonaparte's nephew, Louis-Napoleon, who declared the Second Empire in 1852. Pierre-Joseph "Property is theft!" Proudhon, perhaps the first to adopt the term anarchist, said of the outcome of the Revolutions of 1848: "We have been beaten and humiliated... scattered, imprisoned, disarmed and gagged. The fate of European democracy has slipped from our hands." Breunig, Charles (1977), The Age of Revolution and Reaction, 1789 – 1850
The Sicilian revolt was short lived. The Hungarian revolution ended in a compromise in 1867 that established the Austro-Hungarian Empire. Switzerland had a brief civil war that ended in the establishment of a federal state.
In Germany, the working class and the middle class turned on each other, and were largely crushed by conservative, ruling classes. It would not be until the Franco-Prussian War in 1871 that Prussia united the states under its vision of Kleindeutschland.
In Italy, the risorgimento or resurgence lasted for decades following the Sicilian revolution of 1848, until Italy was unified with Rome as its capital in 1871. Great powers such as Austria and France continued to intervene politically and militarily throughout.
But the big takeaway from the movements of 1848 was this: they gave birth to a whole new era of competing ideas and ideologies.
Marx published The Communist Manifesto in February of 1848, but had little overall impact on those events. Rather, it became a intellectual touchstone for ideological movements to come. It was after 1848 that socialism began to branch and specialize. Anarchist, socialist, communist and libertarian were almost meaningless, undefined terms in 1848. Prior to that, you might speak of Hegelians, or Proudhonists, or Fourierists. Then came the First International, and the ideological battles between Mikhail Bakunin and Karl Marx over the role of the state and the individual. Both philosophies have shaped the motivations behind everything from the authoritarian Bolshevik Revolution in Russia in 1917 to the anti-authoritarian protest movements behind the Battle of Seattle and Occupy Wall Street.
Early Italian nationalists such as Giuseppe Mazzini, who was bitterly critical of Karl Marx, paved the way for ideas which influenced the unification of Italy and Germany, and were still being echoed as the right to popular self-determination even to our present. These movements effectively laid the philosophical and intellectual framework of the next 200 years.
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By 1968, do you mean the Prague Spring and its aftermath, or the Hippie movement in the US?
If Prague, then: Both movements failed. The only revolution in 1848 to hold out for more than a couple of months was the Hungarian revolution, and that was ultimately crushed.