The differences between investigations and inquiries
When running workshops for experienced maths teachers, I hear the claim that Inquiry Maths is just another name for investigations. On one occasion, a teacher appeared exasperated as she accused me of "re-inventing the wheel" and declared that "we've come full circle in maths teaching." Evidently, I had failed to distinguish between investigations and inquiries, but, more importantly, I had also failed to understand that the colleague remembered a time when the investigation classroom was very different to what we know of it today.
The general conception of an investigation has undergone a change in the UK since the National Curriculum came into force over 20 years ago, and particularly since 2001 when the National Numeracy Strategy (NNS) became a statutory requirement in maths classrooms. If teachers used investigations in the NNS era, they were required to meet lesson objectives and fit them into three-part lessons. Investigations had to be structured and sequenced by the teacher to ensure the whole class reached the required outcome. 
The reason the investigation survived at all during this period, it could be argued, was its inclusion in the GCSE specification as coursework. However, coursework investigations became so structured by the requirements of the exam board's mark scheme that they fell into disrepute. No longer was a piece of coursework a reflection of a student’s independent thinking, but rather a reflection of how well the teacher knew the mark scheme. Coursework was scrapped in 2007.   
Investigation classrooms were not always like this. During my PGCE year in the early 1990s, I visited a maths department that taught the whole curriculum through investigations. It was one of the last, if not the very last, school in the country to do so. It was like no other department I have visited or worked in since. Students were allowed to investigate or not, depending on whether the teacher’s questions about a starting point had aroused their curiosity. They investigated individually, occasionally having discussions with the teacher, until they discovered the mathematical concept for which the starting point had been designed. There was no whole-class instruction, which seems unbelievable today.
This is how I imagine the classroom Marion Bird describes in her 1983 booklet on generating mathematical activity. Marion refers to the activities she uses as 'inquiries' and I would class some of them as inquiries in the sense I use the word today. Splitting decominoes, for example, starts with a diagram and, even though Marion sets an initial question, she allows the activity to develop into multiple pathways that encompass different forms of mathematical reasoning. However, another one of her activities - The greatest number of intersections - has become a classic investigation in which students have to draw more diagrams, tabulate results, identify a pattern and discover the generalisation. This is the inductive method of a science experiment in which more results confirm the hypothesis or lead to its revision. 
The exclusively inductive approach is not consistent with the combination of induction and deduction that characterises mathematics, and certainly not with the deductive nature of mathematical proof. Polya explains in How to Solve It and Mathematics and Plausible Reasoning how "deduction completes induction." While the mathematician finds an interesting result through plausible, experimental, and provisional reasoning, the result of this creative work is established definitively with a rigorous proof. It is the certitude given by a proof that makes further results unnecessary. (I have discussed the limitations of the inductive approach in an article from 2008.) The table below summarises the differences between investigations and inquiries as I see them.
  Investigation Inquiry
Starting pointTeacher’s questions to direct students from a starting point towards the target knowledge.Students’ questions, observations or conjectures about a prompt.                  
Regulation of thinking and activityIndividual; occasional discussion with teacher.Social and justified publicly; evaluated by teacher.
Forms of 
Inductive (empirical): tabulate results, identify patterns, generalise from particular cases.Inductive and deductive: exploration, plausible inference, proof.
AimIndividual student’s discovery of a mathematical concept.Independent inquirers who can regulate the process by which maths is developed.
Teacher's role Facilitate the discovery by using (leading) questions in discussions with individuals.Give instruction on concepts and forms of reasoning when appropriate or requested to do so.

Andrew Blair
August 2013 (updated June 2016)

An example: Adding fractions
The differences between investigations and inquiries are exemplified by contrasting a SMILE resource sheet from the 1990s to the Adding fractions inquiry on this site. The first part of the SMILE sheet (below) requires students to find patterns by extending the columns. The dots and blanks act as an invitation to fill in the missing fractions. We notice that the denominator of the first fraction has been chosen to ensure the sum of the fractions cannot be simplified. We also notice that students do not need to know how to add fractions to find the rule; they can infer from the examples that the sum and product of the denominators give you, respectively, the numerator and denominator on the right-hand side. Students would be expected to check their results by verifying that they follow the pattern or comply with the new-found rule.
The second part of the sheet (below) induces students to revise their rule. It sets up a cognitive conflict (in Piaget's terminology) that requires the child to adapt to the new situation by acknowledging that the existing rule does not work. The new rule for finding the numerator in the solution becomes: "For both fractions, multiply the numerator by the denominator in the other fraction and add the two results." The rule is an effective procedure, but there does not seem to be any expectation that the student knows why it works. The end point of the investigation, towards which the teacher might guide the student if necessary, is the discovery of the general rule. 
In contrast, the inquiry starts with just two equations (right). Unlike the dots and blanks on the SMILE sheet, there is no clear path set out for students to follow. In the initial phase of making observations and posing questions about the prompt, students might spontaneously look for a pattern through an inductive process. However, the suggestions that arise are more difficult to verify with only two equations to work with. For example, it could be argued that the next equation equals 9/18 or 9/20. When the rule for adding and multiplying the denominators does arise, the fewer confirming cases make it more speculative than in the investigation.
The uncertainty in both the direction and content of the inquiry raises a question that students do not face in the investigation: How should the class proceed? In the Inquiry Maths model students have a mechanism (the regulatory cards) through which they participate in answering the question. Importantly, they can request an explanation of adding fractions from the teacher or another student. The procedure (or rule), which was the outcome of the investigation, is used as a psychological tool (in Vygotsky's terminology) to facilitate students’ calculations during the inquiry.
The inquiry aims to confirm students’ plausible inferences through deductive reasoning. To that end the teacher might introduce or co-construct an algebraic form of the generalisation (such as, a/b + c/d = (ad + bc)/bd), which acts as a precursor of a deductive proof that the two sides are equal. Furthermore, the inquiry process encourages students to change the prompt. Considering other operations and different types of fractions leads students to enrich their understanding of calculations with fractions and the relationship between different types of calculations. The aim of the investigation is to find a rule through individual discovery; the aim of the inquiry is to achieve a class-wide conceptual understanding of using operations with fractions.