What are the main technical differences between accurately calling somatic point mutations vs copy number variation (CNV) in exome data? Side note: would you need other -omic data to accurately infer CNV (exome data isn't enough)?
Exome data can be used to determine both somatic point mutations and copy number variations. The limiting element for the power of detection for both is read count (how many reads per gene).
Exome data: Exome data uses a reference (for a trio, that would be parents; for cancer, that would be a paired normal sample from blood or non-invasive tissue) in order to determine if a certain point is mutated or not. Next-generation sequencing is prone to sequencing errors, so having a high enough read count is essential to properly calling a somatic mutation. Even so, issues with highly repetitive sites can make this difficult.
Using exome data for copy number is happening a lot nowadays. The number of reads is used as a proxy for how many copies exist, and the balance of alleles at any given gene tells you whether there has been loss of heterozygosity. Of course, with exome data, you aren't getting truly genomic data. But there is enough coverage using the exome to pick up the big events.
CNV data: Copy number analysis is now conventionally done using a high density SNP (single nucleotide polymorphism) array. These are designed to be biallelic, and come in two types. First are the copy number SNPs, which are designed to give the best information about copy number. Then there are the genotyping SNPs, which give information about loss of heterozygosity. Often, SNP array data is higher quality, but this is determined by your platform. Newer platforms have extremely high coverage of the genome and can allow you to really pick out focal changes.
These arrays, based on manufacturer's recommendations, are not meant to be used for somatic mutation calling.
As far as bioinformatic pipelines, you can use similar tools for each. For example, the Aroma package in R has pipelines for both Exome & SNP array data. GISTIC 2.0 from Broad can be used to find focal copy number changes.
A word of caution: It is always best to validate what you find in one platform with another. I recently worked on a paper that used both Exome sequencing and SNP array data to determine copy number. Using GISTIC, there were no focal peaks that matched. The broad chromosomal trends were the same, and that's what the group reported.