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Transcription is the first step in gene expression. It involves copying a gene's DNA sequence to make an RNA molecule. Transcription is performed by enzymes called RNA polymerases, which link nucleotides to form an RNA strand (using a DNA strand as a template).

Specifically, gene expression is controlled on two levels. First, transcription is controlled by limiting the amount of mRNA that is produced from a particular gene. The second level of control is through post-transcriptional events that regulate the translation of mRNA into proteins.

For most intents and purpose TPM is superior to CPM (at least for short read RNA-seq; where the length of a given transcript affects the number of reads produced from said transcript). However, CPM should give comparable results to using TPM. In your case, I would try both and compare the results. Use TPM.

When you use TPM, the sum of all TPMs in each sample are the same. This makes it easier to compare the proportion of reads that mapped to a gene in each sample. In contrast, with RPKM and FPKM, the sum of the normalized reads in each sample may be different, and this makes it harder to compare samples directly.

When you use TPM, the sum of all TPMs in each sample are the same. This makes it easier to compare the proportion of reads that mapped to a gene in each sample. In contrast, with RPKM and FPKM, the sum of the normalized reads in each sample may be different, and this makes it harder to compare samples directly.

By definition, TPM and RPKM are proportional. However, TPM is unit-less, and it additionally fulfils the invariant average criterion. For a given RNA sample, if you were to sequence one million full-length transcripts, a TPM value represents the number of transcripts you would have seen for a given gene or isoform.

CPM is basically depth-normalized counts, whereas TPM is length-normalized (and then normalized by the length-normalized values of the other genes). If one has to choose between those two choices one typically chooses TPM for most things, since generally the length normalization is handy.