What is historical criticism (also known as higher criticism)? What is textual criticism (also known as lower criticism)? How can they help us understand the scriptures? How can they interefere with our understanding of the scriptures?
For starters, none of the terms are very descriptive of what these disciplines actually set out to do. Lower criticism can be seen as the study of what lies below the text and Higher criticism can be seen as the study of what lies above the text. In practical terms, lower criticism looks at the range of extant manuscripts and other evidence and attempts to reconstruct the content of the text from those clues. It a discipline that has application in non-Biblical texts (Josephus presents many interesting problems, for instance), but due to the exceptional number of manuscripts available to us it's flowered in New Testament studies.
Higher criticism tries to put the text in the historical context in which it was formed. Is the text objective or does it distort its subject because of some vested interest? What can we learn about the culture and people who produced the text? How does the text fit with the other texts from the same time and place? An example of a non-Biblical higher criticism problem is whether or not we know what Socrates actually taught.
The other terms are synonymous, but have slightly different connotations. Historical criticism examines the history of a text before it reached final form and Textual criticism examines the history of a text as its transmitted (by hand). If the former terms signify something like a stack, the later terms signify a timeline:
Historical -> Text -> Textual
So Historical criticism begins with the study of oral tradition (Form criticism), earlier written sources, if any, (Source criticism), and editorial decisions (Redaction criticism). If we expand the timeline, it looks something like this:
Form -> Source -> Redaction -> Text -> Textual
Textual critics often use the pre-history of the text and higher criticism to make decisions about which of the available variants are most likely given equivocal manuscript evidence.
Now you might notice that so far we haven't talked much about the text itself or what it means. Part of the reason is that much of the best evidence for both types of criticism is data that does not relate to the meaning of the text. For instance, historical criticism's crowning achievement is the discovery of Q which can be obtained with a bit of set theory:
Q = (Matthew ∩ Luke)\Mark
Quite a bit of textual criticism is simply an application of statistical analysis.
But the other reason is that both criticism techniques are intertwined with interpretation in ways that aren't simple to pull apart. Like echoes in a wave tank, perturbations in one discipline have ripple effects in all directions. The meaning of a text is partially derived from its historical context and often the best information about the historical context comes from the text in question. Which of several textual variations is judged most authentic can be a product of the reconstruction of the culture in which the text was created.
Discoveries in historical and textual criticism have massive positive impacts on the interpretation of the texts they examine. Modern views of Mark have done much to restore the Gospel as an equal with the other three, for instance. Careful examination of Biblical manuscripts have resulted in surprising and profound insights into the very earliest accounts of Jesus. The scholarship represented by these approaches to hermeneuteics ought not to be forgotten or dismissed.
The number one problem with both higher and lower criticism, in my opinion, comes from the temptation to build castles in the clouds. Since meaning is not the primary framework for these disciplines, their conclusions may cease to be related to what the author(s) of the text intended to convey. There's a risk that, as N. T. Wright comments, "the quest for the divine turns out to be a quest for self-discovery".
Jesus told us the source of his teaching:
Joh 14:24 ... and the word which ye hear is not mine, but the Father’s which sent me.
The Old Testament is the word of God, and Jesus "spoke the dark sayings of old."
And he told his disciples to teach others the same thing:
Mt 28:20 Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you alway, [even] unto the end of the world. Amen.
He said that the Holy Ghost would remind them of everything he taught them:
Joh 14:26 But the Comforter, [which is] the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.
Since the source of the New Testament is made clear by the New Testament itself, higher and lower criticism imterfere with our understanding of scripture by throwing doubt on God's word. Sensus plenior presumes that the source of Jesus's knowledge of himself is the Old Testament, and an active dialog with the father through "dinner theater" similar to those of the prophets. As such it is a type of source criticism based in faith rather than disbelief.
Jesus saw a prophecy that when he was 12 he was to "rebel" and so he stayed in the temple to teach as a way of rebelling without sin. The story of the 9 kings of Chedloamer was his source. He could say, "I only do what I see the Father doing in the scriptures".