I want to replace a sequence name in fasta file with another name

Question 1

Consider using Bio::SeqIO to parse your Fasta dataset, instead of doing it yourself. Bio::SeqIO lives for this task, and is well developed for it. Additionally, if you're in bioinformatics, it would serve you well to get to know Bio::SeqIO. Given this, consider the following:

use strict;
use warnings;
use Bio::SeqIO;

open my $fh, '<', 'annot.txt' or die $!;
my %annot = map { /(\S+)\s+(.+)/; $1 => $2 } <$fh>;
close $fh;

my $in = Bio::SeqIO->new( -file => 'goat300.fasta', -format => 'Fasta' );

while ( my $seq = $in->next_seq() ) {
    my $seqID = $annot{ $seq->id } // $seq->id;
    print "$seqID\n" . $seq->seq . "\n";
}

Output on your datasets:

tumor susceptibility
CCCTTTCTCCCTTCCCAACATTCAGAGATACTGAATCGAAACTCTTACTGTCTGTTAGATGACAAAGAGTTATCCATCCTACATACTCCAATTTCCTTCCGCAACTTGTGATTTCGCCGCTTGAATCTTGACGCCGTGCGTCCACAGTTTGTTGTGTTTTATCAATCAAGGTCATTATCAACCGAAGACGCTATCTATTTTCTTGGCGAAGCTCTCGGAAAGGAGCCATCGAAATGGAAGTATTTCTCAAGAAAGTCCGCGAGTTATCCCGGAAGCAGTTC
ubiquitin ligase e3a
GACCTATACCGGACCGTCACTGAAAGNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNACGATCCAGGCATGGAGTTGTGGTGACGAGTAGGAGGGTCACCGTGGTGAGCGGGAAGCCTCGGGCGTGAGCCTGGGTGGAGCCGCCACGGGTGCAGATCTTGGTGGTAGTAGCAAATATTCAAGTGAGAACCTTGAAGGCCGAGGTGGAGAAGGNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNTCATTTGTATCGCCCGGAAAACGTCACAAGAACGGGAGTTGCGTACAGAA
serine threonine-protein phosphatase 4 catalytic subunit
AAGGGACACCGTTGGGTGAGGCGAGCTGCGTTCCTCGAACCATGGCTTCAAAAAGCGACTTAGACCGTCAGATTGAACAGCTCAGGGCCTGCAAGCTCATTACAGAGGATGAGGTTAAGGCACTCTGCGCTAAGGCGCGTGAGATTTTAATTGAAGAGAGTAATGTCCAGTGCGTGGACTCACCTGTCACGGTTTGTGGCGATATCCACGGCCAGTTTTACGACTTGATTGAACTGTTTAAAGTGGGCGGAGATGTTC
bardet-biedl syndrome 2 protein P
TTACTATTTCTGGGCCTTAAGACTGGCTTAGTCGCTTACGACCCTTATAACAATGTAGATGTATATTATAAGGATCTTCCTGATGGTGCTAACGCTATGTTAATTTATTCAAACTCACCGACAAAGGAACAGAATATGCTTTGGCAGGTGGAAACTGTTCGATAATTGGATTGAACGACGGCGGATGCGAGGTATTTTGGACAGTCACTGGCGACTCCGTTTGCTCTCTTTGCTCGATTAAATCCGACAGCGATAAGTCAAGAGATTTTGTGGTTGGCTCTGAAGATTTTGACATCCGAATCTTCCATGGGGATGCCATAATATATGAAATCACGGAGTCTGATG
succinyl- ligase
AAGAGAAGAGGTGAGTTTGAGTATTGTTTGTGTGTGTGTGGTTGGGTGAGTGTGTGGTATGTGGTGTATGTGTGTGATGAATGTATGTGAAAGAGAGTGATGAATCTCATGGATATGTTCGAGTTCGTGGTTTCCATTGATCGGTTATAGCCGAGATGATGGATGTGTTCCATGTGTCTGATTTCAGTTTAGGATTGTGTTGATGATGTTGATGATGAAAATTGTTGATGGTGATGACGATAGTGATGATGATGACGATGTTTCGGATAATGGTGATGATGATGATGGTTCCGACGATGATGTTTCGCTTGATGATGGTGATAATGATGACTCCGAAAATAACGTTGACTCGGATGAG

The hash %annot is initialized by reading and capturing the contents of your annot.txt data. A Bio::SeqIO object is created using your goat300.fasta file data. The while loop iterates through your fasta sequences. The variable $seqID either takes the associated value of the key in the %annot hash or it keeps the current sequence ID (the // notation means defined or, so that insures $seqID will be defined). Finally, the Fasta record is printed.

Hope this helps!

Question 2

There were a lot of warnings in your code, and your approach was inefficient. Let me first show you a working Perl program. I'll explain afterwards.

#!/usr/bin/perl
use strict;
use warnings;

# Read the annotations file
print"Enter annotated file...\n";
# my $f1 = <STDIN>;
my $f1 = 'annot.txt';
open(my $fh_annotations, '<', $f1) or die "Can't open $f1";
my @annotfile = <$fh_annotations>;
close $fh_annotations;

# Read the sequence file
print"Enter sequence file...\n";
# my $f2 = <STDIN>;
my $f2 = 'goat300.fasta';
open(my $fh_genes, '<', $f2) or die "Can't open $f2";
my @seqfile = <$fh_genes>;
close $fh_genes;

# Process the annotations data
my %names; # this hash is going to hold the names
foreach my $line (@annotfile) {
  chomp $line;                      # remove newline
  my @fields = split /\t/, $line;   # split into array
  $names{$fields[0]} = $fields[1];  # save in the hash as key->value pair
}

# Process the sequence data
foreach my $line (@seqfile) {
  # Look at each line
  if ($line =~ m/>(.+)$/) {
    # If there is a heading there, remember it...
    if (exists $names{$1}) {
      # ... check if we know a name for it and replace it in the line
      $line =~ s/($1)/$names{$1}/;
    }
  }
  # output the line (this would be done to another filehandle)
  print $line;
}

This reads both files and saves them in memory, just like yours did. But instead of trying to build two arrays for the names, I went with a hash, which is a key/value pair. Think of it like an array with names instead of numbers and no particular sorting.

Once these names are set up, I can process the sequence file. I simply look at each line and check if there is a heading there, by looking for the > sign. If it's there (it goes into $1 because of the parenthesis), I look if we have a hash entry (with exists) in our %names hash. If we do, we can replace the heading with the proper name.

After that, we could write it out to a new file. I'm just printing it.

I've used a few other techniques. Unfortunately the literature people get in a BioPerl context is quite outdated. Please take this advice, it will make your live easier.

Always use strict and warnings. They will tell you about problems with your code.
Always declare your variables with my. This is not like other languages, where you need to set up a variable at the top of your problem. You can declare it where you need it. The vars only live in a certain scope, which means between the nearest enclosing { and } brackets, or block.
Use three-argument open and lexical file handles for security. Read more here.
Perl offers foreach as an alternative to the C for loop. In this case, it made things a lot easier.

One more thing about this program: While this example data was rather short, I believe your actual data might be a lot larger. Consider processing the sequence file while you read it so you do not run out of memory. There's no need to save all the lines, unless you want to do something else with them.

open my $fh_out, '>', $filename_out or die $!;
open my $fh_in, '<', $filename_in or die $!;
while (my $line = <$fh_in>) {
  # do stuff with the line, like your regex
  print $fh_out $line;
}
close $fh_in;
close $fh_out;