What is the historical and theological basis for Protestant anti-Catholicism?

The Thirty Years War was a really bad time, and in many ways may be responsible for the grief between Catholics and Protestants. While I would argue this animosity is way, way less bad than it was in the past, in the past it could be very bad.

In Pilgrim's Progress, for example, the Protestant John Bunyan (who was in jail for preaching!), compared the Pope to a Pagan heretic, and calls him dying but not yet dead:

In this light, therefore, he came to the end of the valley. Now I saw in my dream, that at the end of the valley lay blood, bones, ashes, and mangled bodies of men, even of pilgrims that had gone this way formerly; and while I was musing what should be the reason, I espied a little before me a cave, where two giants, Pope and Pagan, dwelt in old times; by whose power and tyranny the men whose bones, blood, ashes, etc., lay there, were cruelly put to death. But by this place Christian went without much danger, whereat I somewhat wondered; but I have learnt since, that Pagan has been dead many a day; and as for the other, though he be yet alive, he is, by reason of age, and also of the many shrewd brushes that he met with in his younger days, grown so crazy and stiff in his joints that he can now do little more than sit in his cave’s mouth, grinning at pilgrims as they go by, and biting his nails because he cannot come at them

No less a luminary than Charles Spurgeon warned against letting down the guard against "Popishness", saying in his sermon The Attractions of Popistry:

Now, all this has mainly passed away, and we are relaxing our resistance against the dreaded foe just in proportion as he grows more formidable. It has become the fashion to condemn controversy and to affect the widest charity for this and all other foes of Christ and of souls. High Presbyterian authority even is quoted as saying, that henceforth our concern with Romanism should be chiefly ironical!

And, this book On the Evils of Popery calls on protestants to "resist the evil aggressions of Rome" in its very first paragraph.

From a Protestant perspective, there is an historical case and a theological case.

In brief, historically, many good Christians died at the hands of the Roman church. Whether it be Hugenots, Waldensians, and many of the Reformers themselves, there is no doubt the church persecuted its theological enemies. That said, there is no doubt the Protestant church in England did the same to its adversaries, and the pattern is not unique to either Catholicism in particular or Christianity in general. (Indeed, one only need examine the Shia-Sunni wars in Islam to see this is a human tendency, not a Christian one.)

Theologically, there are differences between Catholics and Protestants - chiefly:

1. Governance. In general, Protestants have a disdain not for any particular Pope, but for the idea of a Pope in general.

2. Marian Devotion. In general, Catholics have a regard for Mary that strikes many Protestants as worship

3. Latria. What Catholics view as honoring Saints and their icons, many Protestants view as idol worship

4. Purgatory, Indulgences, and other Accretions. Many Protestants view Catholics as having added a lot to Scripture - Purgatory being one example of a concept not really known to Scripture.

Issues like these contributed to a situation in which Protestants had legitimate grievances against a church. Indeed, in much the same was as Anti-Semitism was considered proper for a long time, so too Anti-Catholicism was as American as Apple Pie, leading to the formation of the Freemasons, the KKK, and the Know-Nothings, and derailing the Presidential ambitions of Al Smith in 1928 and nearly costing JFK the Presidency in 1960. (Only by declaring independence from the Pope could he diffuse the issue.)

From a Roman Catholic perspective, Protestants were schismatics - Christians who were denying the authority of the church. This is treason at best and apostasy at worst - serious stuff.

Now, in fairness, I would argue most enlightened Christians on either side now view many of thee issues as secondary, and most of the fighting as historic hurt and not present issues - but to gloss over it is to forget a important divide. These are rarely first order concerns however. We both believe in Jesus as being divine, fully god-fully man, all of the creeds and all that stuff. Indeed, much like Common law, the Protestant position is to accept Catholic dogma, unless if there is a modern precedent to overturn it.

To say there are no grievances between the churches is thus wrong - but it is really easy to overemphasize them too. Christians are still humans. Humans always have something to fight about. So, the answer to your question, Why do Protestants not like Catholics is this:

We're still human.